Question

practice assignment 9.3 double - angle and half - angle identities
score: 2/6 answered: 2/6
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question 3
use a half angle or reduction formula to fill in the blanks in the identity below:
$(\cos(7x))^{2}=\square+\square\cos(\square x)$
question help: worked example 1

Explanation:

Step1: Recall the double - angle formula for cosine

The double - angle formula for cosine is \(\cos(2\alpha)=2\cos^{2}\alpha - 1\). We can rewrite it to solve for \(\cos^{2}\alpha\) as \(\cos^{2}\alpha=\frac{1 + \cos(2\alpha)}{2}\).

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

Let \(\alpha=7x\). Then \(\cos^{2}(7x)=\frac{1+\cos(2\times7x)}{2}\).
Simplify \(\frac{1+\cos(14x)}{2}\) to \(\frac{1}{2}+\frac{1}{2}\cos(14x)\).

Answer:

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