Mathematics Trigonometry questions from JEE Main 2004.
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 $\cos \frac{\alpha-\beta}{2}$ is