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

If $\tan \theta \cdot \tan 2 \theta = 1,$ then the value of $\sin ^{2} 2 \theta + \tan ^{2} 2 \theta$ is equal to

If $A + B + C = 180^\circ$,then $\sum \tan \frac{A}{2} \tan \frac{B}{2} = $

If angle $\theta$ is divided into two parts such that the tangent of one part is $k$ times the tangent of the other and $\phi$ is their difference,then $\sin \theta = $

Given the equation $4x^2 + 4(a - 1)x + (1 - 2a) = 0$ has roots $\sin \theta$ and $\cos \theta$ where $0 < \theta < \frac{\pi}{2}$,then the maximum value of $(a + \sin \theta)$ is:

$\sin^4 \frac{\pi}{8} + \sin^4 \frac{3\pi}{8} + \sin^4 \frac{5\pi}{8} + \sin^4 \frac{7\pi}{8} = $