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Chapter 5: Problem 30

Use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. $$\tan ^{2} 2 x \cos ^{4} 2 x$$

Short Answer

Expert verified
The expression \( \tan ^{2} 2 x \cos ^{4} 2 x \) rewritten in terms of the first power of the cosine using power-reducing formulas is \( \frac{{(1 - \cos 2x)^2}}{{(1 + \cos 2x)^2}} \)

Step by step solution

01

Identify the Power Reducing formulas

To solve the problem, you need to know the power-reducing formulas. The power-reducing formula for cosine is \[ \cos ^{2} x=\frac{1+\cos 2 x}{2} \]
02

Rewrite Tangent

Rewrite the tangent function in terms of sine and cosine \[ \tan ^{2} x = \frac{{\sin ^{2}x}}{{\cos ^{2}x}} \]
03

Apply the Power Reducing formulas

Substitute \[ \cos ^{2} x \] and \[ \sin ^{2} x \] in the formula with their respective power-reducing formulas: \[ \tan ^{2} x \cos ^{4} x = \frac{{(1-\cos 2 x)^{2}/{2}}}{(1+\cos 2 x)^{2}/{2}} \]
04

Simplify the expression

Simplify the expression obtained in step 3 to obtain the answer in terms of the first power of the cosine. \[ \frac{{(1 - \cos 2x)^2}}{2} \cdot \frac{2}{(1 + \cos 2x)^2} = \frac{{(1 - \cos 2x)^2}}{{(1 + \cos 2x)^2}} \]

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