Problem 92 - Algebra and Trigonometry

Question

Without showing algebraic details, describe in words how to reduce the power of \(\cos ^{4} x\).

Step-by-Step Solution

Answer

\(\cos^{4}x = \frac{3}{8} + \frac{1}{2} \cos(2x) + \frac{1}{8} \cos(4x)\)

Step 1: Understand the power-reduction formula

The power-reducing formula for \(\cos^{2}x\) is \(\cos^{2}x = \frac{1+ \cos(2x)}{2}\). This is an identity in trigonometry used for reducing the power of 2 of a cosine function into a simple equation involving the cosine of twice the angle.

Step 2: Apply the power-reduction formula to \(\cos^{4}x\)

To express \(\cos^{4}x\) in terms of \(\cos(2x)\) and \(\cos(4x)\), start by understanding that \(\cos^{4}x = (\cos^{2}x)^2\). Then, substitute the power-reducing formula of \(\cos^{2}x\) into this equation, which gives \((\frac{1+ \cos(2x)}{2})^2\). Then, square this to finally get \(\frac{1}{4} + \frac{1}{2} \cos(2x) + \frac{1}{4} \cos^2(2x)\).

Step 3: Apply the power-reduction formula again

Now, the term with \(\cos^2(2x)\) needs to be reduced further. Applying the power-reduction formula to this term yields \(\cos^{4}x = \frac{1}{4} + \frac{1}{2} \cos(2x) + \frac{1}{4}( \frac{1+ \cos(4x)}{2} )\). The final expression for \(\cos^{4}x\) in terms of \(\cos(2x)\) and \(\cos(4x)\) is \(\cos^{4}x = \frac{3}{8} + \frac{1}{2} \cos(2x) + \frac{1}{8} \cos(4x)\).