If $\sin A + \cos A = \sqrt{2}$,then $\cos^2 A = $

If $\theta$ is an acute angle and $\tan(4\theta - 50^{\circ}) = \cot(50^{circ} - \theta)$,then the value of $\theta$ in degrees is:

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 = $

If $\operatorname{cosec} \theta - \cot \theta = p$,then the value of $\operatorname{cosec} \theta =$

If $\sin \theta + \cos \theta = x,$ then ${\sin ^6}\theta + {\cos ^6}\theta = \frac{1}{4}[4 - 3{({x^2} - 1)^2}]$ for

If $\sin x + \cos x = a$,$a \in [-\sqrt{2}, \sqrt{2}] - \{-1, 1\}$,then $\sum_{n=1}^\infty (\sin^n x + \cos^n x)$ is equal to -