Question

Prove the identity:
(sin⁡(2x-y)+sin⁡y)/(cos⁡(2x-y)+cos⁡y )=tan⁡x

Answer 1

Recall your sum/difference formulas. You have

sin(2x-y)+sin(y) = 2sin(x)cos(x+y)
cos(2x-y)+cos(y) = 2cos(x)cos(x+y)

Now just divide and you have tan(x)