Question

How do you prove $\cos 2x=1-2{\sin }^{2}x$ using other trigonometric identities?

Answer 1

Proof of given relation:

$\cos 2x=1-2{\sin }^{2}x$

Here we have $LHS=\cos 2x$

We know the indentity,

$\cos 2x=\cos \left(x+x\right)$ $\left[\because cos\left(A+B\right)=cosAcosB-sinAsinB\right]$

$\cos 2x={\cos }^{2}x-{\sin }^{2}x$

$=\cos x\cos x-\sin x\sin x$

$={\cos }^{2}x-{\sin }^{2}x$

$=\left(1-{\sin }^{2}x\right)-{\sin }^{2}x$ $\left[\because si{n}^{2}x+co{s}^{2}x=1\right]$

$=1-2{\sin }^{2}x$

$=RHS$

Hence, $\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathbf{2}\mathit{x}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{2}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{x}$ using other trigonometric identities is proved.