1. ## [SOLVED] Trig Identities

Hi there

Ive been trying out this identity;
Prove:

Sinē Θ =1/2 (1- Cos 2Θ)

I got to
Sinē Θ = 1-Cos 2Θ/ 2

but i am unsure about what to do to get out of the next problem, can you cancel the 2???

Could you show how you continue?

This is one of the basic trig identities.

What facts and identities are we allowed to use?

Prove: .$\displaystyle \sin^2\!\theta \:=\:\frac{1}{2}(1-\cos2\theta)$

The right side is: .$\displaystyle \frac{1-\cos2\theta}{2}$ .[1]

If we are allowed the identity: .$\displaystyle \cos(A+B) \:=\:\cos A\cos B - \sin A\sin B$

. . then we have: .$\displaystyle \cos(\theta + \theta) \:=\:\cos\theta\cos\theta - \sin\theta\sin\theta \quad\Rightarrow\quad\cos2\theta \;=\;\cos^2\!\theta - \sin^2\!\theta$

Since $\displaystyle \cos^2\!\theta \:=\:1-\sin^2\!\theta$, we have: .$\displaystyle \cos2\theta \:=\:(1-\sin^2\!\theta) - \sin^2\!\theta \quad\Rightarrow\quad\cos2\theta \:=\:1 - 2\sin^2\!\theta$

Substitute into [1]: .$\displaystyle \frac{1 - (1-2\sin^2\!\theta)}{2}\;=\;\frac{2\sin^2\!\theta}{2} \;=\;\sin^2\!\theta$