Question

Prove that $\frac{sinx-siny}{cosx+cos y}=tan \frac{x-y}{2}.$

Answer 1

It is known that

$sinA-sinB=2cos(\frac{A+B}{2})sin(\frac{A-B}{2}),cosA+cosB=2cos(\frac{A+B}{2})cos(\frac{A-B}{2})$

$∴L.H.S. =\frac{sin\,x-sin\,y}{cos\,x+cos\,y}$

$=\frac{2cos(\frac{x+y}{2}).sin(\frac{x-y}{2})}{2cos(\frac{x+y}{2}).cos(\frac{x-y}{2})}$

$=\frac{sin(\frac{x-y}{2})}{cos(\frac{x-y}{2})}$

$=tan(\frac{x-y}{2})=R.H.S.$