[tex]\bf \textit{Double Angle Identities}
\\ \quad \\
sin(2\theta)=2sin(\theta)cos(\theta)
\\ \quad \\
cos(2\theta)=
\begin{cases}
cos^2(\theta)-sin^2(\theta)\\
1-2sin^2(\theta)\\
\boxed{2cos^2(\theta)-1}
\end{cases}
\\ \quad \\\\ tan(2\theta)=\cfrac{2tan(\theta)}{1-tan^2(\theta)}\\\\
-------------------------------\\\\
[/tex]
[tex]\bf sec(\theta)=\cfrac{csc^2(\theta)}{2cot^2(\theta)-csc^2(\theta)}\\\\
-------------------------------\\\\
\textit{doing the right-hand-side}
\\\\\\
\cfrac{\frac{1}{sin^2(\theta)}}{\frac{2cos^2(\theta)}{sin^2(\theta)}-\frac{1}{sin^2(\theta)}}\implies \cfrac{\frac{1}{sin^2(\theta)}}{\frac{2cos^2(\theta)-1}{sin^2(\theta)}}\implies \cfrac{1}{sin^2(\theta)}\cdot \cfrac{sin^2(\theta)}{2cos^2(\theta)-1}
\\\\\\
\cfrac{1}{2cos^2(\theta)-1}\implies \cfrac{1}{cos(2\theta)}\implies sec(2\theta)[/tex]
