Let:
[tex]\begin{gathered} x+3y=7_{\text{ }}(1) \\ 5x-6y=-28_{\text{ }}(2) \end{gathered}[/tex]From (1):
[tex]x=7-3y_{\text{ }}(3)[/tex]Replace (3) into (2):
[tex]\begin{gathered} 5(7-3y)-6y=-28 \\ 35-15y-6y=-28 \\ 35-21y=-28 \\ -21y=-63 \\ y=\frac{-63}{-21} \\ y=3 \end{gathered}[/tex]Replace the value of y into (3):
[tex]x=7-3(3)=7-9=-2[/tex]Therefore:
x = -2
y = 3
or
(x,y) = (-2,3)
Accrotime is a manufacturer of quartz crystal watches. Accrotime researchers have shown that the watches have an average life of 30 months before certain electronic components deteriorate, causing the watch to become unreliable. The standard deviation of watch lifetimes is 4 months, and the distribution of lifetimes is normal.
A. a) If Accrotime guarantees a full refund on any defective watch for 2 years after purchase, what percentage of total production will the company expect to replace? (Round your answer to two decimal places.)\
B. If Accrotime does not want to make refunds on more than 6% of the watches it makes, how long should the guarantee period be (to the nearest month
a) The percentage of total production will the company expect to replace is of 6.62%.
b) The guarantee period should be of 23 months.
Normal Probability DistributionThe z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, depending if the obtained z-score is positive or negative.Using the z-score table, the p-value associated with the calculated z-score is found, and it represents the percentile of the measure X in the distribution.The mean and the standard deviation for the duration of the watches are given as follows:
[tex]\mu = 30, \sigma = 4[/tex]
The proportion of watches that will be replaced are those that last less than 2 years = 24 months, hence it is given by the p-value of Z when X = 24, as follows:
Z = (24 - 30)/4
Z = -1.5.
Z = -1.5 has a p-value of 0.0662.
Hence the percentage is of 6.62%.
For item b, the guarantee period should be the 6th percentile, which is X when Z = -1.555, hence:
-1.555 = (X - 30)/4
X - 30 = -1.555 x 4
X = 23 months.
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Hi, can you help me answer this question please, thank you!
Given that
[tex]\begin{gathered} \mu_1=sample\text{ of soda in the coke can} \\ \mu_2=sample\text{ of soda in the pepsi can} \end{gathered}[/tex]Therefore, in the first statement, we are to test how accurate the companies package these cans.
Mathematically it can be expressed as,
[tex]H_0\colon\mu_1\leq\mu_2[/tex]In the second statement, we wish to test the claim that the mean of the amount of liquid in coke cans is greater than the amount of liquid in pepsi cans. This can be expressed mathematically as,
[tex]H_a\colon\mu_1>\mu_2[/tex]Hence, the correct option is Option 1.
A phone company charges for service according to the formula: C(n) = 19 + 0.08n , where n is the number of minutes talked , and C(n) is the monthly charge, in dollars .The rate of change in this equation is:The initial value in this equation is:
In a linear equation, the coefficient of the variable is the rate of change and the constant term is the initial value.
For example, a linear function with rate of change m and initial value b is:
[tex]f(x)=mx+b[/tex]In the given formula, the variable is n, its coefficient is 0.08 and the constant term is 19.
Since n is measured in minutes and C is measured in dollars (as well as the initial value), then, the coefficient 0.08 must have the units necessary for the minutes to cancel out, leaving dollars as the unit of 0.08n. Then, the units of the rate of change must be dollars per minute.
Since the constant term is 19, then the initial value is 19.
Therefore, the answers are:
The rate of change in the equation is 0.08 dollars per minute.
The initial value in the equation is 19 dollars.
Triangle A'B'C' is apparently - у А A' B C С B' O A clockwise 90 degree rotation of Triangle ABC O A reflection across the y-axis of Triangle ABC O A translation of Triangle ABC right 7 units O A clockwise 270 degree rotation of Triangle ABC
Since all coordinates of the transformated triangle are changed like this:
[tex](x,y)\rightarrow(y,-x)[/tex]Triangle A'B'C' is a clockwise 270 degree rotation of triangle ABC
A counterclockwise rotation of 90º is the same that a clockwise rotation of 270º
The surface area of a rectangular prism is 60. Which of the following are possible dimensions of the rectangular prism? 1 Pt
The surface area of a rectangular prism with sides a, b and c is given by the formula:
[tex]S=2(ab+ac+bc)[/tex]Use that formula to find the surface area that a prism with the given dimensions on each option would have.
A. 6, 2, 1 1/2
[tex]\begin{gathered} S=2(6\cdot2+6\cdot1\frac{1}{2}+2\cdot1\frac{1}{2}) \\ =2(12+9+3) \\ =2(24) \\ =48 \end{gathered}[/tex]B. 5, 4, 1 1/4
[tex]\begin{gathered} S=2(5\cdot4+5\cdot1\frac{1}{4}+4\cdot1\frac{1}{4}) \\ =2(20+\frac{25}{4}+5) \\ =2(\frac{125}{4}) \\ =\frac{125}{2} \\ =62.5 \end{gathered}[/tex]C. 3, 4, 1 1/2
[tex]\begin{gathered} S=2(3\cdot4+3\cdot1\frac{1}{2}+4\cdot1\frac{1}{2}) \\ =2(12+3\frac{3}{2}+6) \\ =2(21\frac{3}{2}) \\ =45 \end{gathered}[/tex]D. 6, 3, 1 1/3
[tex]\begin{gathered} S=2(6\cdot3+6\cdot1\frac{1}{3}+3\cdot1\frac{1}{3}) \\ =2(18+8+4) \\ =2(30) \\ =60 \end{gathered}[/tex]Therefore, the only possible dimensions of a rectangular prism with surface area 60 listed on the options, are 6, 3 and 3 1/3. The answer is:
[tex]\text{Option D}[/tex]What is this expression in simplest form?
+2
4x² + 5x + 1
Ο Α.
О в.
O C.
O D.
4x + 1
²-4
(x + 1)(x − 2)
-
I
(z = 2)
1
4x + 1
(x + 1)(x-2)
ww
4x+1
#12
Answer:
[tex] \frac{x + 2}{4 {x}^{2} + 5x + 1 } . \frac{4x + 1}{ {x}^{2} + 4} \\ \frac{x + 2}{4 {x}^{2} + (4 + 1)x + 1 } . \frac{4x + 1}{ {(x + 2)}^{2} } \\ \frac{1}{4 {x}^{2} + 4x + x + 1}. \frac{4x + 1}{x + 2} \\ \frac{1}{x(4x + 1) + 1(4x + 1)} . \frac{4x + 1}{x + 2} \\ \frac{1}{(4x + 1)(x + 1)} . \frac{4x + 1}{x + 2} \\ \frac{1}{(x + 1)}. \frac{1}{(x + 2)} \\ \frac{1}{(x + 1)(x + 2)} [/tex]
A. is the answer!!
The value of 0.36 when converted to a fraction in the simplest form is 9/25.
What is fraction?A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters.
here, we have,
to calculate fractions in simplest form:
Your information is incomplete. Therefore, an overview will be given. It should be noted that a fraction is in its simplest form when the numerator and denominator are prime.
From example, let's convert 0.36 to a fraction on its simplest form. This will be:
0.36 = 36/100 = 9/25
In conclusion, 0.36 is 9/25 in the simplest form.
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Draw the image of the figure under thegiven transformation.6.reflection across the x-axis7. (X,y) - (x - 4, y + 1)8. reflection across the y-axis
6.
While transformation with the reflection across the x-axis, the absicssa (x-coordinate) remains the same but ordinate (y-coordinate) changes its sign.
The coordinate of point A is (3,0), coordinate of point B is (1,4) and the coordinate of point C is (5,3).
After transformation the, coordinate with the image can be shown as,
Thus, the coordinates of the image after transformation is A'(3,0), B'(1,-4) and C'(5,-3).
A blueprint of a shopping complex shows the bottom edge of the roof to be 68 feet above the ground. If the roof rises to a point 122 feet above the ground over a horizontal distance of 4.5 yards, what is the slope of the roof?41.2128
SOLUTION
Given:
A blueprint of a shopping complex shows the bottom edge of the roof to be 68 feet above the ground. If the roof rises to a point 122 feet above the ground over a horizontal distance of 4.5 yards, what is the slope of the roof?
[tex]Slope\text{ =}\frac{rise}{run}[/tex][tex]\begin{gathered} Rise\text{ =122-68=54 feet} \\ Run=4.5yards=13.5feet \end{gathered}[/tex][tex]\therefore Slope=\frac{54}{13.5}=4feet[/tex]
Final answer:
Let’s say that the patient drinks an 8-ounce glass of orange juice. The nurse must convert that to milliliters (mL) or cubic centimeters (cc) in order to record the intake volume in the patient’s fluid input and output chart. (Remember 1 mL = 1 cc.)
Answer: 8 ounce = 237 ml
Explanation:
From the information given,
volume of drink taken by patient = 8 ounce
We want to convert 8 ounce to ml
Recall,
1 US fluid ounce = 29.57 ml
Thus,
8 ounce = 8 x 29.57 = 236.56
Approximately,
8 ounce = 237 ml
Justin earned $600 last week fixing computers.Is it possible to determine how many hours Justin worked?explain
Since Justin earned $600 last week
If we want to find the number of hours that he worked, we must have how much he earned per hour
But we do not have how much did he earn per hour, so
It is impossible to find how many hours did he work from the given information
The answer is
No, it is impossible to find that
Select all inputs for which f (x)=2A:x=-7B:x=0C:x=4D:none of the above
To answer this question, we need to see the graph of the function carefully.
For x = -7, we can see that the function is equal to 2, that is:
[tex]f(-7)=2[/tex]For x = 0, we have that the function is equal to -1, that is:
[tex]f(0)=-1[/tex]For x = 4, we have that the function is equal to -2, that is:
[tex]f(4)=-2[/tex]We need to find the value of x. Then, we have to find the point where we "touch" the function, and then find the y-value of the function.
Therefore, in summary, we have that the only input for which f(x) = 2 is when x = -7 (option A).
Determine the height of the ball after 4 seconds *Height of Ball over Time160144128112Height (in feet)96NO643216O1 2 3 4 5 6Time (in seconds)
80 ft
1) In the graph, we can find out the height after 4 seconds simply locating the point when the x-axis at t=4
2) Hence, after 4 seconds the height of the ball is:
3) Hence, the answer is 80 ft (the blue dot) (4,80)
Consider functions f and g.1 + 12f(1) = 12 + 4. – 12for * # 2 and 7 -64.2 – 16. + 1641 +48for a # -12 Which expression is equal to f(x) · g(t)?OA.41 - 81 + 61OB.SIKIAOC.21 + 6I + 2D.6
Given the following functions below,
[tex]\begin{gathered} f(x)=\frac{x+12}{x^2+4x-12}\text{ and} \\ g(x)=\frac{4x^2-16x+16}{4x+48} \end{gathered}[/tex]Factorising the denominators of both functions,
Factorising the denominator of f(x),
[tex]\begin{gathered} f(x)=\frac{x+12}{x^2+4x-12}=\frac{x+12}{x^2+6x-2x-12}=\frac{x+12}{x(x+6)-2(x+6)}=\frac{x+12}{(x-2)(x+6)} \\ f(x)=\frac{x+12}{(x-2)(x+6)} \end{gathered}[/tex]Factorising the denominator of g(x),
[tex]\begin{gathered} g(x)=\frac{4x^2-16x+16}{4x+48}=\frac{4(x^2-4x+4)}{4(x+12)} \\ \text{Cancel out 4 from both numerator and denominator} \\ g(x)=\frac{x^2-4x+4}{x+12}=\frac{x^2-2x-2x+4}{x+12}=\frac{x(x-2)-2(x-2)}{x+12}=\frac{(x-2)^2}{x+12} \\ g(x)=\frac{(x-2)^2}{x+12} \end{gathered}[/tex]Multiplying both functions,
[tex]undefined[/tex]What is the surface area of a pyramid if its one triangular face has an area of 20 sq. cm and its square base has a side of 5cm?A. 100 sq.cmB. 105 sq.cmC. 110 sq. cmD. 115 sq.cm
To solve this problem, we will use the following formula for the surface area of a square pyramid:
[tex]SA=BA+4A_t,[/tex]where BA= base area, and A_t is the area of one of the lateral triangles.
Now, we are given that:
[tex]A_t=20cm^2.[/tex]Recall that the area of a square is given by the following formula:
[tex]A_s=side^2.[/tex]We are given that the side of the base is 5 cm long, therefore, the base area is:
[tex]A_s=(5cm)^2=25cm^2.[/tex]Finally, we get that:
[tex]SA=25cm^2+4(20cm^2)=105cm^2.[/tex]Answer:
[tex]105cm^2.[/tex]+10B-10-8-644810Bc8--10and aWe can show that ABC is congruent to AA'BC by a translation ofunits)across the-axis.
ANSWER
ABC is congruent to A'B'C' by a translation of 2 units and a reflection across the x-axis
EXPLANATION
We want to identify the transformations that were carried out on ABC to obtain A'B'C'.
First, we notice that A'B'C' is not aligned with ABC because the vertex of B' is 2 units behind the vertex of B.
So, we can say that there was a translation of 2 units to the left or -2 units..
Also, we notice that the vertices of ABC were flipped over the x-axis to obtain A'B'C'.
Therefore, we can conclude that there was a reflection across the x axis.
Hence, ABC is congruent to A'B'C' by a translation of 2 units and a reflection across the x-axis.
Given that: 100 = 2^2 * 5^2, how is 400 written as a product of its prime factors?
Answer:
400 written as a product of it's prime factors is 2^4 * 5²
Step-by-step explanation:
We have to factore the number, dividing by prime factors(2, 3, 5, 7, ...)
400|2
200|2
100
We already have the factorization of 100. So
400 = 2²*100 = 2²*2²*5² = 2^4 * 5²
400 written as a product of it's prime factors is 2^4 * 5²
A box contains different colored paper clips. The probability of drawing two red paper clips from the box without replacement is 1/7, and the probability of drawing one red paper clip is 2/5.What is the probability of drawing a second red paper clip, given that the first paper clip is red?1/65/142/32/35
SOLUTION
The probability of two red paper clips without replacement means
The probability of drawing the first and the probability of drawing the second one. This is represented as
[tex]P(R_{1st}\cap R_{2nd})[/tex]And this was given as
[tex]\frac{1}{7}[/tex]So
[tex]P(R_{1st}\cap R_{2nd})=\frac{1}{7}[/tex]Probalilty of drawing one red paper clip is
[tex]P(R_{1st})=\frac{2}{5}[/tex]Now the probability of drawing a second red paper clip, given that the first paper clip is red becomes
[tex]\begin{gathered} P(R_{1st}/R_{2nd})=\frac{P(R_{1st}\cap R_{2nd})}{P(R_{1st})} \\ \\ P(R_{1st}/R_{2nd})=\frac{\frac{1}{7}}{\frac{2}{5}} \\ \\ =\frac{5}{14} \end{gathered}[/tex]Simplify the following expression by combining liker terms: 4x +9 -6x +610X-1510x+15-2x+152x+15
We have the expression 4x+9-6x+6 and we have to simplify it.
We can group the terms that are alike. We have two groups of terms: the ones that are multiplied by "x" and the ones that are just numbers alone.
Then, the terms 4x and (-6x) are alike and we can add them, getting (-2x). In this terms, x is the common denominator of both terms.
The other terms that are alike are 9 and 6, that can be added to get 15.
[tex]\begin{gathered} 4x+9-6x+6 \\ (4-6)x+(9+6) \\ -2x+15 \end{gathered}[/tex]The answer is -2x+15.
katie has 5.455 apples and sadie has 10.31 how many apples do they have in all
Katie has 5.455
Sadie has 10.31
To find how many apples they have in all add the two numbers
They have = 5.455 + 10.31
They have = 5.455 + 10.310
0 + 5 = 5
1+ 5 = 6
3 + 4 = 7
10 + 5 = 15
They have = 15.765 apples in all
If AABC = ADEC,ZB = 44º and ZE = 4xx= [?]
Solution
Given that Triangle, ABC is congruent to Triangle DEC
=> ∠A = ∠D; ∠B = ∠E; ∠C = ∠C
Given that ∠B = 44, ∠E = 4x
=> 44 = 4x
=>x = 44/4 = 11
Hence, x = 11
Write the equation in slope-intercept form through the point (2, -1) and is perpendicular to the line y = -5x + 1 and graph.
First, we are going to calculate the perpendicular slope. The condition for perpendicular lines is the following:
[tex]m1m2=-1[/tex]First, m1 = -5
[tex]m2=\frac{-1}{m1}=\frac{-1}{-5}\rightarrow m2=\frac{1}{5}[/tex]Now, for b
[tex]b=y-m2x[/tex]For the point (2,-1)
[tex]b=-1-\frac{1}{5}\cdot(2)[/tex][tex]b=-\frac{5}{5}-\frac{2}{5}=\frac{-7}{5}[/tex][tex]y=\frac{1}{5}x-\frac{7}{5}[/tex]A hot air balloon is flying above Groveburg. To the left side of the balloon, the balloonist measure the angle of depressionto the Groveburg soccer fields to be 20° 15'. To the right side of the balloon, the balloonist measures the angle ofdepression to the high school football field to be 62° 30'. The distance between the two athletic complexes is 4 miles.What is the distance from the balloon to the football field?a.b.>3.6 miC.~6.2 mi>2.2midy1.4 miPlease select the best answer from the choices providedOAOBOCOD
The distance from the balloon to the football field will be 1.4 miles.
Angle of depression to the Grove burg soccer fields = 20° 15'.
Use 1' = 1 / 60° :
15' = 1 / 4 ° = 0.25 °
20° 15' = 20.25°
Angle of depression to the high school football field = 62° 30'.
30' = 0.5°
62° 30' = 62.5°
the distance from the balloon to the football field will be:
Let the distance be a
a / sin a = c / sin c
a / sin (20.25) = 4 / sin (97.5)
a = 4 sin (20.25) / sin (97.5)
a = 1.4 miles.
Therefore, we get that, the distance from the balloon to the football field will be 1.4 miles.
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in a survey of 1300 people who owned a certain type of car 585 said they would buy the type of car again what percent of people surveyed were satisfied with the car
In order to find the percentage of people that were satisfied with the car, we just need to divide this amount of people by the total amount of people surveyed.
So we have:
[tex]\frac{585}{1300}=0.45=45\text{\%}[/tex]So 45% of the surveyed people were satisfied with the car.
Write the expression and simplifyThe difference of -10 and the product of p and q
We start with a subtraction, where we want to subtract the second term from - 10. The second term consists in a multiplication between p and q. Writing this as a mathematical expression we have
[tex]-10-pq[/tex]This expression is already on simplest form.
The lengths of the diagonals of a rectangle are representedby 2x + 3 and 4x - 11. Find the value of x.
The problem says that there are formulas for the diagonals of a rectangle. they are both givan in terms of x
Now, recall that the diagonals of a rectangle should be equal to each other, therefore those two expressions must equal ech other
2 x + 3 = 4 x - 11
Now let's solve for x:
move all terms with "x" to one side of the equation, and all "numerical" terms to the other side:
start by subtracting 2x from both sides:
3 = 4 x - 2 x - 11
3 = 2 x - 11
Now add 11 to both sides:
11 + 3 = 2 x
14 = 2 x
then x is 7
Write the equation of the line that is perpendicular to the line given and through the given point. Do not use spaces in your equation. y=-2X+1 (0,5) *
Answer:
y = 0.5x + 5
Explanation:
The equation of a line can be calculated as:
[tex]y=m(x-x_1)+y_1[/tex]Where m is the slope and (x1, y1) is a point in the line.
To find the slope of our line, we need to identify the slope of the given line.
Since the equation of the given line is y = -2x + 1, the slope of this line is -2, because it is the number beside the x.
Then, two lines are perpendicular if the product of their slopes is equal to -1. So, we can write the following equation:
[tex]-2\cdot m=-1[/tex]Therefore, the slope m of our line will be:
[tex]m=\frac{-1}{-2}=0.5[/tex]Now, we can replace the value f m by 0.5 and the point (x1, y1) by (0, 5) and we get that the equation of the line is:
[tex]\begin{gathered} y=0.5(x-0)+5 \\ y=0.5(x)+5 \\ y=0.5x+5 \end{gathered}[/tex]Therefore, the answer is y = 0.5x + 5
2) Reflection across the line y = x. B(-2, 1), S(-2, 2), R(3, 3), H(2, -2) B'( ) S/ ) R (2) J( )
Let's begin by identifying key information given to us:
[tex]B\mleft(-2,1\mright),S\mleft(-2,2\mright),R\mleft(3,3\mright),H\mleft(2,-2\mright)[/tex]Reflection is done over the line y = x, we have:
[tex]\begin{gathered} (x,y)\rightarrow(y,x) \\ B\mleft(-2,1\mright)\rightarrow B^{\prime}(1,-2) \\ S\mleft(-2,2\mright)\rightarrow S^{\prime}(2,-2)_{} \\ R\mleft(3,3\mright)\rightarrow R^{\prime}(3,3) \\ H\mleft(2,-2\mright)\rightarrow H^{\prime}(-2,2) \end{gathered}[/tex]a cellular phone company charges a base rate of $15.00 per month and $0.05 per minute,m, which equation could be used to find the total monthly charge in dollars,c?
Suppose you use "m" minutes in a month and each minute is $0.05, so your minute bill would be:
0.05 * m
Per month, there is a fixed rate of $15 , no matter how many minutes you use.
So, that's a fixed cost.
Total monthly charge would be:
15 + 0.05m
The cost, c, is:
[tex]c=0.05m+15[/tex]Correct Answer Choice is Option B
how do u knwo which way to face the inequality sign in the answer of these questions like 1>x<3 how do u know which way to face them. i put some examples os u cna use them to explain
Using the graph identify the intervals:
a) Function being less than or equal to 0: In which x interval is the graph under the x-axis (the functions are less than 0 when they are under x-axis)
As the ineqaulity sing is less than or equal to 0, the interval includes those x-values for which the function is 0:
Solution: Interval from x=1 to x=3
[tex]\begin{gathered} x^2-4x+3\leq0 \\ 1\leq x\leq3 \\ \lbrack1,3\rbrack \end{gathered}[/tex]b) Function being greater than or equal to 0: In which x interval is the graph over the x-axis.
As the ineqaulity sing is greater than or equal to 0, the interval includes those x-values for which the function is 0:
Solution: Interval from - infinite to 1 and from 3 to infinite
[tex]\begin{gathered} x^2-4x+3\ge0 \\ 1\ge x\ge3 \\ (-\infty,1\rbrack\cup\lbrack3,\infty) \end{gathered}[/tex]c) Function being greater than 0: In which x interval is the graph over the x-axis.
As the ineqaulity sing is greater than to 0, the interval does not include those x-values for which the function is 0.
[tex]\begin{gathered} x^2-4x+3>0 \\ 1>x>3 \\ (-\infty,1)\cup(3,\infty) \end{gathered}[/tex]d) Function being less than 0: In which x interval is the graph under the x-axis.
As the ineqaulity sing is less than 0, the interval does not include those x-values for which the function is 0:
[tex]\begin{gathered} x^2-4x+3<0 \\ 1sec40+ sec20 tan2 0 - 2 tan4 0 =3 sec² 0 -2Sect0-210
Given:
[tex]sec^4\theta+sec^2\theta tan^2\theta-2tan^4\theta=3sec^2\theta-2[/tex]Required:
We need to prove the given equation.
Explanation:
Consider the left-hand side of the equation.
[tex]Add\text{ and subtract }3tan^4\theta.[/tex][tex]sec^4\theta+sec^2\theta tan^2\theta-2tan^4\theta=sec^4\theta+sec^2\theta tan^2\theta-2tan^4\theta+3tan^4\theta-3tan^4\theta[/tex][tex]=sec^4\theta+sec^2\theta tan^2\theta+tan^4\theta-3tan^4\theta[/tex][tex]Add\text{ and subtract -2}sec^2\theta tan^2\theta.[/tex][tex]=sec^4\theta+sec^2\theta tan^2\theta+tan^4\theta-3tan^4\theta-2sec^2\theta tan^2\theta+2sec^2\theta tan^2\theta[/tex][tex]=sec^4\theta-2sec^2\theta tan^2\theta+tan^4\theta-3tan^4\theta+sec^2\theta tan^2\theta+2sec^2\theta tan^2\theta[/tex][tex]=sec^4\theta-2sec^2\theta tan^2\theta+tan^4\theta-3tan^4\theta+3sec^2\theta tan^2\theta[/tex][tex]Use\text{ }sec^4\theta-2sec^2\theta tan^2\theta+tan^4\theta=(sec^2\theta-tan^2\theta)^2[/tex][tex]=(sec^2\theta-tan^2\theta)^2-3tan^4\theta+3sec^2\theta tan^2\theta[/tex][tex]=(sec^2\theta-tan^2\theta)^2+3tan^2\theta(sec^2\theta-tan^2\theta)[/tex][tex]Use\text{ }sec^2\theta-tan^2\theta=1.[/tex][tex]=1^2+3tan^2\theta(1)[/tex][tex]=1+3tan^2\theta[/tex][tex]Use\text{ }tan^2\theta=sec^2\theta-1.[/tex][tex]=1+3(sec^2\theta-1)[/tex][tex]=1+3sec^2\theta-3[/tex][tex]=3sec^2\theta-2[/tex]We get the right-hand side of the equation.
Final answer:
[tex]sec^4\theta+sec^2\theta tan^2\theta-2tan^4\theta=3sec^2\theta-2[/tex]