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Chapter 5: Problem 27

Use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. $$\cos ^{4} x$$

Short Answer

Expert verified
Using the power-reducing formula, \(\cos ^{4} x\) can be rewritten as \((\frac{1 + cos(2x)}{2})^2\)

Step by step solution

01

Recall the power-reducing formula

The power-reducing formula for cosine is given by \(cos^{2}(x) = \frac{1 + cos(2x)}{2}\)
02

Apply the power-reducing formula for \(\cos ^{2} x\)

Replace \(\cos ^{2} x\) with \(\frac{1 + cos(2x)}{2}\)
03

Apply power-reducing formula for \(\cos ^{4} x\)

As given that \(\cos ^{4} x = (cos^{2}(x))^2\), you can substitute what you obtained from step 2 into this formula to calculate the \(\cos ^{4} x\) formula. Hence, \(\cos ^{4} x = (\frac{1 + cos(2x)}{2})^2\)

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Most popular questions from this chapter

Use the half-angle formulas to simplify the expression. $$\sqrt{\frac{1+\cos 4 x}{2}}$$

Use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. $$\sin ^{2} 2 x \cos ^{2} 2 x$$

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$7 \pi / 12$$

Use the product-to-sum formulas to rewrite the product as a sum or difference. $$7 \cos (-5 \beta) \sin 3 \beta$$

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\cos (x+\pi)-\cos x-1=0$$
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