Problem 112 - Algebra and Trigonometry

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 simplified expression is: \((\frac{1}{2} - \frac{1}{2} \cos(2x))^3\) or \( \frac{1}{8} - \frac{3}{8}\cos(2x) + \frac{3}{8}\cos^2(2x) - \frac{1}{8}\cos^3(2x)\)

Step 1: Relate given power to square power

The given power is 6, which is a multiple of 2. So, we can write \(\sin ^6 x\) as \((\sin ^2 x)^3\) = \(\sin ^2 x \sin ^2 x \sin ^2 x\).

Step 2: Apply power-reducing formula

The power-reducing formula is \(\sin ^2 x = \frac{1}{2} - \frac{1}{2} \cos(2x)\). So, replace every occurrence of \(\sin ^2 x\) in our expression with this formula.

Step 3: Simplify the resulting expression

Carry out the multiplication and simplify the expression, keeping in mind the rule \(\cos^2 (x) = 1 - \sin^2 (x)\).