Question

Please prove the following identity

Answer 1

Let's remember the factorizations

`a^3 + b^3 = (a+b)(a^2-ab+b^2)`

`a^3- b^3=(a-b)(a^2+ab+b^2)`

Now

`(\sin ^3 \theta + \cos^3 \theta)/(\sin \theta + \cos \theta) + (\sin^3 \theta - \cos^3 \theta)/(\sin \theta - \cos \theta)`

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

`=\sin ^2 \theta - \sin \theta\cos \theta + \cos^2 \theta + \sin^2 \theta + \sin \theta \cos \theta + \cos^2 \theta`

`=\sin ^2 \theta + \cos^2 \theta + \sin^2 \theta+ \cos^2 \theta`

`=2 \quad\checkmark`