Answer
Three forms of the double angle formula for $\cos 2\theta $ are:
$\underline{\cos 2\theta ={{\cos }^{2}}\theta -{{\sin }^{2}}\theta }$, $\underline{\cos 2\theta =1-2{{\sin }^{2}}\theta }$ and $\cos 2\theta =2{{\cos }^{2}}\theta -1$
Work Step by Step
We know that the double angle formula gives the relationship between angle, $\theta,$ and angle, $2\theta .$
In the case of cosine, the double angle formula can be obtained from the formula for cosine of the sum of angles. The formula for cosine of the sum of angles is:
$\cos \left( \alpha +\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta $
Therefore,
$\cos 2\theta =\cos \left( \theta +\theta \right)$
$\cos \left( \theta +\theta \right)$ is similar to $\cos \left( \alpha +\beta \right)$ with $\alpha =\beta =\theta .$ So,
$\begin{align}
& \cos \left( 2\theta \right)=\cos \left( \theta +\theta \right) \\
& =\cos \theta \cos \theta -\sin \theta \sin \theta \\
& ={{\cos }^{2}}\theta -{{\sin }^{2}}\theta
\end{align}$
So
$\cos 2\theta ={{\cos }^{2}}\theta -{{\sin }^{2}}\theta .$
Now, from the Pythagorean identity,
$\begin{align}
& {{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1 \\
& {{\cos }^{2}}\theta =1-{{\sin }^{2}}\theta
\end{align}$
Apply $\left( {{\cos }^{2}}\theta =1-{{\sin }^{2}}\theta \right)$ in $\left( \cos 2\theta ={{\cos }^{2}}\theta -{{\sin }^{2}}\theta \right)$
After that,
$\begin{align}
& \cos 2\theta ={{\cos }^{2}}\theta -{{\sin }^{2}}\theta \\
& =\left( 1-{{\sin }^{2}}\theta \right)-{{\sin }^{2}}\theta \\
& =1-2{{\sin }^{2}}\theta
\end{align}$
So,
$\cos 2\theta =1-2{{\sin }^{2}}\theta $
Similarly,
$\cos 2\theta ={{\cos }^{2}}\theta -{{\sin }^{2}}\theta .$
Also, from the Pythagorean identity,
$\begin{align}
& {{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1 \\
& {{\sin }^{2}}\theta =1-{{\cos }^{2}}\theta
\end{align}$
Apply $\left( {{\sin }^{2}}\theta =1-{{\cos }^{2}}\theta \right)$ in $\left( \cos 2\theta ={{\cos }^{2}}\theta -{{\sin }^{2}}\theta \right)$
Then,
$\begin{align}
& \cos 2\theta ={{\cos }^{2}}\theta -{{\sin }^{2}}\theta \\
& ={{\cos }^{2}}\theta -\left( 1-{{\cos }^{2}}\theta \right) \\
& =-1+2{{\cos }^{2}}\theta
\end{align}$
Thus,
$\cos 2\theta =2{{\cos }^{2}}\theta -1$
