$\sin^6 \theta + \cos^6 \theta + 3 \sin^2 \theta \cos^2 \theta = $

If $\tan A + \cot A = 2$,then the value of $\tan^{4} A + \cot^{4} A$ is:

If $a$ and $b$ respectively represent the lengths of a side and a diagonal of a regular pentagon that is inscribed in a circle, then $\frac{b}{a}=$

$\operatorname{Sech}^{-1}(\sin \alpha) =$

Let $\alpha$ and $\beta$ be two real roots of the equation $(k+1) \tan^{2} x - \sqrt{2} \lambda \tan x = (1-k)$,where $k(\neq -1)$ and $\lambda$ are real numbers. If $\tan^{2}(\alpha+\beta) = 50$,then a value of $\lambda$ is:

The number of all possible values of $\theta$,where $0 < \theta < \pi$,for which the system of equations
$(y+z) \cos 3\theta = (xyz) \sin 3\theta$
$x \sin 3\theta = \frac{2 \cos 3\theta}{y} + \frac{2 \sin 3\theta}{z}$
$(xyz) \sin 3\theta = (y+2z) \cos 3\theta + y \sin 3\theta$
have a solution $(x_0, y_0, z_0)$ with $y_0 z_0 \neq 0$,is