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Chapter 5: Problem 51

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

Short Answer

Expert verified
Rewriting \( \sin^{4}x \cos^{2}x \) using power-reducing formulas gives the result \( \frac{1}{16} - \frac{1}{8}\cos (2x) + \frac{1}{32} + \frac{1}{32} \cos(4x) \)

Step by step solution

01

Identify the applicable power-reducing formulas

The power-reducing formulas are \( \cos ^{2} x=\frac{1+\cos 2 x}{2} \) and \( \sin ^{2} x=\frac{1-\cos 2 x}{2} \). These formulas can be used to reduce the powers of cosine and sine in the expression.
02

Substitute using the power-reducing formulas

Substitute \( \cos ^{2} x \) and \( \sin ^{4} x \) in the expression \( \sin ^{4} x \cos ^{2} x \) using the power-reducing formulas. This results in \( \left(\frac{1-\cos 2 x}{2}\right)^{2} \times \frac{1+\cos 2 x}{2} \).
03

Simplify the expression

Expand and simplify the expression \( \left(\frac{1-\cos 2 x}{2}\right)^{2} \times \frac{1+\cos 2 x}{2} \) which gives you \( \frac{1-2\cos 2x +\cos ^{4} 2x}{16} \times (1+ \cos 2x) \). This simplifies further to \( \frac{1}{16} - \frac{1}{8}\cos 2x + \frac{1}{16}\cos ^{2} 2x \). This can be further simplified using the power reducing formula \( \cos ^{2} 2x = \frac{1+\cos 4x}{2} \). Substituting gives a final expression of \( \frac{1}{16} - \frac{1}{8}\cos 2x + \frac{1}{32} + \frac{1}{32}\cos 4x \).

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