$\cos \left(\frac{3 \pi}{4}+x\right)-\sin \left(\frac{\pi}{4}-x\right) = $

If $\cos \theta = \frac{3}{5}$ and $\cos \phi = \frac{4}{5},$ where $\theta$ and $\phi$ are positive acute angles,then $\cos \frac{\theta - \phi}{2} = $

Prove that: $2 \cos \frac{\pi}{13} \cos \frac{9 \pi}{13} + \cos \frac{3 \pi}{13} + \cos \frac{5 \pi}{13} = 0$

$\frac{\sin(B + A) + \cos(B - A)}{\sin(B - A) + \cos(B + A)} = $

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 . . . . . .

For $\alpha, \beta \in \left(0, \frac{\pi}{2}\right)$,let $3 \sin (\alpha+\beta)=2 \sin (\alpha-\beta)$ and a real number $k$ be such that $\tan \alpha=k \tan \beta$. Then the value of $k$ is equal to :