Question

Verify that (1+cos2theta)/(costheta)=2costheta is an identity.

Answer 1

$\bf \textit{Double Angle Identities} \\\\ cos(2\theta)= \begin{cases} cos^2(\theta)-sin^2(\theta)\\ 1-2sin^2(\theta)\\ 2cos^2(\theta)-1&\leftarrow \textit{we'll use this one} \end{cases}\\\\[-0.35em] \rule{31em}{0.25pt}\\\\ \cfrac{1+cos(2\theta )}{cos(\theta )}=2cos(\theta )\\\\[-0.35em] ~\dotfill\\\\ \cfrac{1+[2cos^2(\theta )-1]}{cos(\theta )}\implies \cfrac{2cos^2(\theta )}{cos(\theta )}\implies \cfrac{2\underline{cos(\theta )} cos(\theta )}{\underline{cos(\theta )}}\implies 2cos(\theta )$