How do you verify Cos 2x= 2 sin (x+pi/4) cos (x+pi/4)?

1 Answer
Feb 9, 2016

Explanation given below.

Explanation:

Verify $\cos \left(2 x\right) = 2 \sin \left(x + \frac{\pi}{4}\right) \cos \left(x + \frac{\pi}{4}\right)$

First let us understand two identities

color(Blue)(sin(2theta)=2sin(theta)cos(theta)

color(Blue)(sin(pi/2 + theta) =cos(theta)

Now let us take our problem

$\cos \left(2 x\right) = 2 \sin \left(x + \frac{\pi}{4}\right) \cos \left(x + \frac{\pi}{4}\right)$

Let us start with the Right Hand side

$= 2 \sin \left(x + \frac{\pi}{4}\right) \cos \left(x + \frac{\pi}{4}\right)$

=sin(2(x+pi/4) double angle formula for sine.

$= \sin \left(2 x + \frac{\pi}{2}\right)$

$= \cos \left(2 x\right) \quad$ by the second identity shared above.