Question

use a half angle formula or formula for reducing powers to fill in the blanks in the identity below:

$$(\cos(7x))^{2}=\square+\square\cos(\square x)$$

Explanation:

Step1: Recall the double - angle formula for cosine

The double - angle formula for cosine is \(\cos(2\alpha)=2\cos^{2}\alpha - 1\). Rearranging this formula to solve for \(\cos^{2}\alpha\), we get \(\cos^{2}\alpha=\frac{1 + \cos(2\alpha)}{2}\).

Step2: Substitute \(\alpha = 7x\) into the formula

Let \(\alpha=7x\). Then \((\cos(7x))^{2}=\frac{1+\cos(2\times7x)}{2}\).

Step3: Simplify the expression

$$
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$$

Answer:

\((\cos(7x))^{2}=\frac{1}{2}+\frac{1}{2}\cos(14x)\)