Answer
The required value of cos2A is $Co{{s}^{2}}A-{{\sin }^{2}}A,\,2Co{{s}^{2}}A-1$ and $1-2{{\sin }^{2}}A$.
Work Step by Step
In order to find the value of cos 2A, the double angle formula is used that is as shown below:
$\cos \left( \alpha +\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta $
Now, consider $\alpha $ and $\beta $ as x in cos (A+A), then the formula becomes:
$\begin{align}
& \cos \,2A=\cos \left( A+A \right) \\
& =\cos \,A\,\cos \,A-\sin \,A\,\sin \,A \\
& ={{\cos }^{2}}A-{{\sin }^{2}}A
\end{align}$
With the help of the aforementioned formula, the proving can be done as given below:
$\begin{align}
& \cos 2A={{\cos }^{2}}A-{{\sin }^{2}}A \\
& ={{\cos }^{2}}A-\left( 1-{{\cos }^{2}}A \right) \\
& ={{\cos }^{2}}A-1+{{\cos }^{2}}A \\
& =2{{\cos }^{2}}A-1
\end{align}$
The third formula for cos 2A is as shown below:
$\begin{align}
& \cos 2A={{\cos }^{2}}A-{{\sin }^{2}}A \\
& =1-{{\sin }^{2}}A-{{\sin }^{2}}A \\
& =1-2{{\sin }^{2}}A
\end{align}$
Hence, the required value of cos2A is $Co{{s}^{2}}A-{{\sin }^{2}}A,\,2Co{{s}^{2}}A-1$ and $1-2{{\sin }^{2}}A$.
