If $\sin \theta + \operatorname{cosec} \theta = 2$,then the value of $\sin^{10} \theta + \operatorname{cosec}^{10} \theta$ is equal to

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 $\alpha, \beta$ are acute angles such that $\sin \beta=2 \sin \alpha$ and $3 \cos \beta=2 \cos \alpha$,then $\sec (\alpha+\beta)=$

Evaluate: $\sin 6^{\circ} + \sin 54^{\circ} + \sin 126^{\circ} + \cos 156^{\circ}$

If $\cos \alpha + \cos \beta = a$ and $\sin \alpha + \sin \beta = b$,then match the items given in List-$A$ with those of their values in List-$B$.

List-$A$	List-$B$
$(I)$ $\tan \left(\frac{\alpha + \beta}{2}\right) =$	$(a)$ $\frac{b}{a}$
$(II)$ $\cos (\alpha + \beta) =$	$(b)$ $\frac{2ab}{a^2 + b^2}$
$(III)$ $\sin (\alpha + \beta) =$	$(c)$ $\frac{2ab}{a^2 - b^2}$
$(IV)$ $\tan (\alpha + \beta) =$	$(d)$ $\frac{a^2 - b^2}{a^2 + b^2}$

The value of $\sum_{r=1}^{18} \cos^2(5r)^\circ$,where $x^\circ$ denotes the $x$ degree,is equal to