$\cos 20^\circ \cos 40^\circ \cos 80^\circ = $

If $\tan \alpha = \frac{-12}{5}$,$\cot \beta = \frac{7}{24}$,$\alpha$ does not belong to the second quadrant,and $\beta$ does not belong to the first quadrant,then $\sqrt{13} \sin \frac{\alpha}{2} + \cos \frac{\beta}{2} + \tan \frac{\alpha}{2} \cot \frac{\beta}{2} = $

If $\cosh(x) = \frac{5}{4}$,then $\cosh(3x)$ is equal to

The value of $x$ in $\left(0, \frac{\pi}{2}\right)$ satisfying the equation $(\sin x)(\cos x) = \frac{1}{4}$ is

If $\cos A = \frac{3}{4}$,then $32 \sin \left(\frac{A}{2}\right) \sin \left(\frac{5A}{2}\right) = $

Prove that: $\cos 6x = 32 \cos^6 x - 48 \cos^4 x + 18 \cos^2 x - 1$