question_answer

The expression \[{{\left( \frac{\cos A+\cos B}{\sin A-\sin B} \right)}^{n}}+\left( \frac{\sin A+\sin B}{\cos A-\cos B} \right)=\]

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

Correct Answer: A

Solution :

The given expression \[={{\left( \frac{2\cos \left( \frac{A+B}{2} \right)\cos \left( \frac{A-B}{2} \right)}{2\cos \left( \frac{A+B}{2} \right)\sin \left( \frac{A-B}{2} \right)} \right)}^{n}}\]\[+{{\left( \frac{2sin\left( \frac{A+B}{2} \right)\cos \left( \frac{A-B}{2} \right)}{2sin\left( \frac{A+B}{2} \right)\sin \left( \frac{B-A}{2} \right)} \right)}^{n}}\] \[={{\cot }^{n}}\left( \frac{A-B}{2} \right)+{{(-1)}^{n}}{{\cot }^{n}}\left( \frac{A-B}{2} \right)\]