Question

`( \sin^(2) ( (\pi)/( 4 ) - \alpha ) ) = (1)/(2) (1 - \sin(2 \alpha ) )`

Prove​

Answer 1

Explanation:

`\sin^2 ((\pi)/(4) - \alpha) = (1)/(2)(1 - \sin 2\alpha)`

Use the identity

`\sin^2 \theta = (1 - \cos 2\theta)/(2)`

on the left side.

`(1 - \cos [2((\pi)/(4) - \alpha)])/(2) = (1)/(2)(1 - \sin 2\alpha)`

`(1 - \cos ((\pi)/(2) - 2\alpha))/(2) = (1)/(2)(1 - \sin 2\alpha)`

Now use the identity

`\sin \theta = \cos((\pi)/(2) - \theta)`

on the left side.

`(1 - \sin 2\alpha)/(2) = (1)/(2)(1 - \sin 2\alpha)`

`(1)/(2)(1 - \sin 2\alpha) = (1)/(2)(1 - \sin 2\alpha)`