If $\sin x-\sin y=\frac{27}{65}$ and $\cos x-\cos y=\frac{-21}{65}$, then $\sin (x+y)=$

The number of solutions of the equation $\sec x \cdot \cos 5 x+1=0$ in the interval $[0,2 \pi]$ is

If the equation $2 \cot ^{-1}\left(x^2+2 x+k\right)=\pi-3 \tan ^{-1} \left(x^2+2 x+k\right)$ has two distinct real solutions, then all the values of $k$ lie in the interval

$$ \sec h^{-1}(\sin \alpha)= $$