Problem 112 - Precalculus Essentials

Question

Use the power-reducing formulas to rewrite \(\sin ^{6} x\) as an equivalent expression that does not contain powers of trigonometric functions greater than 1

Step-by-Step Solution

Answer

The equivalent expression that does not contain powers of trigonometric functions greater than 1 is \(\frac{1}{8}(1 - 3\cos(2x) + \frac{3}{2}(1+\cos(4x)) - \cos^3(2x))\).

Step 1: Apply power-reducing formula

First, rewrite \(\sin^6(x)\) as \((\sin^2(x))^3\), which is equivalent. Then apply the power reducing formula for \(\sin^2(x)\), getting \(\left(\frac{1 - \cos(2x)}{2}\right)^3\).

Step 2: Expand the expression

Expand the expression \(\left(\frac{1 - \cos(2x)}{2}\right)^3\) like a normal polynomial. The expanded form will be \(\frac{1}{8}(1 - 3\cos(2x) + 3\cos^2(2x) - \cos^3(2x))\). Since there is no term with power greater than two, further simplification is required.

Step 3: Further Simplification

For the term with power of two, \(\cos^2(2x)\), apply the power-reducing formula for \(\cos^2(x)\) to get \(\frac{1}{2}(1 + \cos(4x))\). Substitute this back into the expression obtained in step 2 to get the final simplified expression as \(\frac{1}{8}(1 - 3\cos(2x) + \frac{3}{2}(1+\cos(4x)) - \cos^3(2x))\).