Question

Prove the identity:
(cosx-cosy/sinx+siny)+(sinx-siny/cosx+cosy)=0

Answer 1

\[\begin{align*}
LHS&=\frac{\cos x-\cos y}{\sin x+\sin y}+\frac{\sin x-\sin y}{\cos x+\cos y}\\\\

&=\cdots\times\frac{\cos x+\cos y}{\cos x+\cos y}+\cdots\times\frac{\sin x+\sin y}{\sin x+\sin y}\\\\

&=\frac{(\cos x-\cos y)(\cos x+\cos y)+(\sin x-\sin y)(\sin x+\sin y)}{(\sin x+\sin y)(\cos x+\cos y)}\\\\
&=\cdots
\end{align*}\]

Answer 2

So do you multiply across the top of the fraction then?

Answer 3

Yes, expand, then use the identity \(\sin^2x+\cos^2x=1\). Everything will work out nicely.

Answer 4

okay, thank you

Answer 5

yw