Question

prove that:cos2A=cos^2 A-sin^2 A

Answer 1

cos 2A = cos ( A + A ) 
cos 2A = cos A cos A - sin A sin A 
cos 2A = cos ² A - sin ² A

Answer 2

To prove that: $\cos 2A$ = $\cos^2 A$ - $\sin^2 A$.

L.H.S. = $\cos 2A$

= $\cos (A+A)$

Using the trigonometric identity:

$\cos (A+B)$ = $\cos A \cos B - \sin A \sin B$

Put B = A, we get

= $\cos A \cos A - \sin A \sin A$

= $\cos^2 A$ - $\sin^2 A$

= R.H.S., proved.

∴ $\cos 2A$ = $\cos^2 A$ - $\sin^2 A$. proved.

Thus, $\cos 2A$ = $\cos^2 A$ - $\sin^2 A$, proved.