question_answer

Let\[\alpha ,\beta \]be such that\[\pi <\alpha -\beta <3\pi .\].If\[\sin \alpha +\sin \beta =-\frac{21}{65}\]and\[\cos \alpha +\cos \beta =-\frac{27}{65},\]then the value of\[\cos \left( \frac{\alpha -\beta }{2} \right)\]is

A) \[-\frac{3}{\sqrt{130}}\]               

B)        \[\frac{3}{\sqrt{130}}\] 

C)        \[\frac{6}{65}\]

D)                        \[-\frac{6}{65}\]

Correct Answer: A

Solution :

Given that,                          ...(i) and                         ...(ii) On squaring and adding Eqs. (i) and (ii), we get