Find the value of $\sin 15^{\circ}$.

If $\tan(A + B) = p$ and $\tan(A - B) = q,$ then the value of $\tan(2A)$ in terms of $p$ and $q$ is

If $\tan A + \tan B = x$ and $\cot A + \cot B = y$,then $\tan (A + B) =$

Let $\cos(\alpha+\beta)=-\frac{1}{10}$ and $\sin(\alpha-\beta)=\frac{3}{8}$ where $0 < \alpha < \frac{\pi}{3}$ and $0 < \beta < \frac{\pi}{4}$. If $\tan 2\alpha=\frac{3(1-r\sqrt{5})}{\sqrt{11}(s+\sqrt{5})}$,where $r, s \in N$,then $r+s$ is equal to . . . . . .

Suppose $\theta_1$ and $\theta_2$ are such that $(\theta_1-\theta_2)$ lies in the $3^{\text{rd}}$ or $4^{\text{th}}$ quadrant. If $\sin \theta_1+\sin \theta_2=-\frac{21}{65}$ and $\cos \theta_1+\cos \theta_2=-\frac{27}{65}$,then $\cos \left(\frac{\theta_1-\theta_2}{2}\right)=$

Prove that $\cos \left( \frac{\pi}{4} + x \right) + \cos \left( \frac{\pi}{4} - x \right) = \sqrt{2} \cos x$.