If $\tan \theta + \cot \theta = 5,$ then $\tan^2 \theta + \cot^2 \theta$ is

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 :

If $A + B + C = \pi ,$ then $\frac{{\cos A}}{{\sin B\sin C}} + \frac{{\cos B}}{{\sin C\sin A}} + \frac{{\cos C}}{{\sin A\sin B}} = $

If $\frac{3\pi}{4} < \alpha < \pi,$ then $\sqrt{\csc^2 \alpha + 2\cot \alpha}$ is equal to

If $\theta$ is in the first quadrant and $\tan \theta = \frac{3}{4},$ then $\frac{\tan \left(\frac{\pi}{2}-\theta\right)-\sin (\pi-\theta)}{\sin \left(\frac{3 \pi}{2}+\theta\right)-\cot (2 \pi-\theta)} = $

$\cos ^{2} 5^{\circ}+\cos ^{2} 10^{\circ}+\cos ^{2} 15^{\circ}+\cdots+\cos ^{2} 90^{\circ}=$