$\tan \alpha + 2\tan 2\alpha + 4\tan 4\alpha + 8\cot \,8\alpha = $

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

The value of $\cos 15^\circ - \sin 15^\circ $ is equal to

$\tan 3A - \tan 2A - \tan A = $

If $\frac{{\cos x}}{a} = \frac{{\cos (x + \theta )}}{b} = \frac{{\cos (x + 2\theta )}}{c} = \frac{{\cos (x + 3\theta )}}{d} \, ,$ then $\left( {\frac{{a + c}}{{b + d}}} \right)$ is equal to :-

Prove that $\frac{\cos 4 x+\cos 3 x+\cos 2 x}{\sin 4 x+\sin 3 x+\sin 2 x}=\cot 3 x$