[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)\\
\boxed{1-2sin^2(\theta)}\\
2cos^2(\theta)-1
\end{cases}
\\ \quad \\\\
tan(2\theta)=\cfrac{2tan(\theta)}{1-tan^2(\theta)}\\\\
-------------------------------[/tex]
[tex]\bf cos(2\theta )+sin(\theta )=0\implies 1-2sin^2(\theta )+sin(\theta )=0
\\\\\\
0=2sin^2(\theta )-sin(\theta )-1\implies 0=[2sin(\theta )+1][sin(\theta )-1]\\\\
-------------------------------\\\\
0=2sin(\theta )+1\implies -1=2sin(\theta )\implies -\cfrac{1}{2}=sin(\theta )
\\\\\\
\measuredangle \theta =
\begin{cases}
\frac{7\pi }{6}\\\\
\frac{11\pi }{6}
\end{cases}\\\\
-------------------------------\\\\
0=sin(\theta )-1\implies 1=sin(\theta )\implies \measuredangle \theta =\frac{\pi }{2}[/tex]
