Question
The expression $${\left( {\frac{{\cos A + \cos B}}{{\sin A - \sin B}}} \right)^n} + {\left( {\frac{{\sin A + \sin B}}{{\cos A - \cos B}}} \right)^n} = $$
A.
$$2\,{\cot ^n}\left( {\frac{{A - B}}{2}} \right)$$ if $$n$$ is even
B.
$$0$$ if $$n$$ is even
C.
$$2\,{\cot ^n}\left( {\frac{{A - B}}{2}} \right)$$ if $$n$$ is odd
D.
$$3$$ if $$n$$ is odd
Answer :
$$2\,{\cot ^n}\left( {\frac{{A - B}}{2}} \right)$$ if $$n$$ is even
Solution :
The given expression
$$\eqalign{
& = {\left( {\frac{{2\cos \left( {\frac{{A + B}}{2}} \right)\cos \left( {\frac{{A - B}}{2}} \right)}}{{2\cos \left( {\frac{{A + B}}{2}} \right)\sin \left( {\frac{{A - B}}{2}} \right)}}} \right)^n} + {\left( {\frac{{2\sin \left( {\frac{{A + B}}{2}} \right)\cos \left( {\frac{{A - B}}{2}} \right)}}{{2\sin \left( {\frac{{A + B}}{2}} \right)\sin \left( {\frac{{B - A}}{2}} \right)}}} \right)^n} \cr
& = {\cot ^n}\left( {\frac{{A - B}}{2}} \right) + {\left( { - 1} \right)^n}{\cot ^n}\left( {\frac{{A - B}}{2}} \right) \cr} $$
\[ = \left\{ \begin{array}{l}
2\,{\cot ^n}\left( {\frac{{A - B}}{2}} \right),{\rm{ if }}\,n\,{\rm{ is\, even}}\\
0,{\rm{ if }}\,n\,{\rm{ is\, odd}}
\end{array} \right.\]