Question

Prove that sinx-siny/cosx+cosy=tan x-y/2

Answer 1

Sinx - Siny /CosX+CosY= tan(X-Y)/2


We know trigonometry sum identities

SinX-SinY= 2Cos (X+Y)/2 × Sin(X-Y)/2

Similarly

CosX+ CosY = 2Cos(X+Y)/2 × Cos(X-Y)/2

putting the value

Now
LHS
$\frac{ \sin(x) - \sin(y) }{ \cos(x) + \cos(y) } = \frac{ \sin( \frac{x - y}{2} ) }{ \cos( \frac{x - y}{2} ) } \\ \\ = \tan( \frac{x - y}{2} )$

R.HS
tan (x-y)/2

LHS=RHS proved

Answer 2

2cos x+y/2 sin x-y/2 / 2cos x+y/2 cos x-y/2 = tan x-y/2