Question

Simplify cos^(2) 6θ − sin^(2) 6θ by using either the double-angle or half-angle formulas

Answer 1

Recall your double-angle identities


$\bf \textit{Double Angle Identities} \\ \quad \\ sin(2\theta)=2sin(\theta)cos(\theta) \\ \quad \\\\ cos(2\theta)= \begin{cases} \boxed{cos^2(\theta)-sin^2(\theta)}\\ 1-2sin^2(\theta)\\ 2cos^2(\theta)-1 \end{cases} \\ \quad \\\\ tan(2\theta)=\cfrac{2tan(\theta)}{1-tan^2(\theta)}\\\\ -----------------------------\\\\ cos^2(6\theta)-sin^2(6\theta)\iff cos[2(6\theta)]\implies cos(12\theta)$