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Chapter 5: Problem 52

Verify each identity. $$\sin ^{4} t-\cos ^{4} t=1-2 \cos ^{2} t$$

Short Answer

Expert verified
The given trigonometric identity \( \sin^{4}t - \cos^{4}t = 1 - 2\cos^{2}t \) holds true.

Step by step solution

01

Recognize the left side as a difference of squares

Rewrite the left-hand side \(\sin^{4}t - \cos^{4}t\) as a difference of squares. It can be written as \((\sin^{2}t + \cos^{2}t)(\sin^{2}t - \cos^{2}t)\).
02

Use the Pythagorean Identity to simplify the first part

Use the Pythagorean Identity \(\sin^{2}t + \cos^{2}t = 1\) to simplify the first part. This gives the expression \(1(\sin^{2}t - \cos^{2}t)\).
03

Write the expression in another form

Express \(\sin^{2}t - \cos^{2}t\) as \(1 - 2\cos^{2}t\). Therefore, the left-hand side of the equation becomes \(1(1-2\cos^{2}t)\), simplifying to \(1-2\cos^{2}t\). This now matches the right-hand side.
04

Confirm that both sides are equal

Now the equation reads \(1 - 2\cos^{2}t = 1 - 2\cos^{2}t\), verifying that the given identity holds true.

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Most popular questions from this chapter

Use this information to solve. When throwing an object, the distance achieved depends on its initial velocity, \(v_{0}\) and the angle above the horizontal at which the object is thrown, \(\theta\) The distance, \(d\), in feet, that describes the range covered is given by $$d=\frac{v_{0}^{2}}{16} \sin \theta \cos \theta$$ where \(v_{0}\) is measured in feet per second. You and your friend are throwing a baseball back and forth. If you throw the ball with an initial velocity of \(v_{0}=90\) feet per second, at what angle of elevation, \(\theta,\) to the nearest degree, should you direct your throw so that it can be easily caught by your friend located 170 feet away?

Describe the difference between verifying a trigonometric identity and solving a trigonometric equation.

Will help you prepare for the material covered in the first section of the next chapter. Solve each equation by using the cross-products principle to clear fractions from the proportion: If \(\frac{a}{b}=\frac{c}{d},\) then \(a d=b c,(b \neq 0 \text { and } d \neq 0)\) Round to the nearest tenth. $$\text { Solve for } a: \frac{a}{\sin 46^{\circ}}=\frac{56}{\sin 63^{\circ}}$$

Will help you prepare for the material covered in the first section of the next chapter. Solve each equation by using the cross-products principle to clear fractions from the proportion: If \(\frac{a}{b}=\frac{c}{d},\) then \(a d=b c,(b \neq 0 \text { and } d \neq 0)\) Round to the nearest tenth. $$\text { Solve for } B, 0

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$\cos x-5=3 \cos x+6$$
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