Questions in trigonometry

Select	Question
	If $\tan (\pi \cos \theta )=\cot (\pi \sin \theta )$, then $\sin \left( \theta +\frac{\pi }{4} \right)$ equals
	The value of $\theta $lying between 0 and $\pi /2$and satisfying the equation $\begin{vmatrix} {1 + {{\sin }^2}\theta }&{{{\cos }^2}\theta }&{4\sin 4\theta }\\ {{{\sin }^2}\theta }&{1 + {{\cos }^2}\theta }&{4\sin 4\theta }\\ {{{\sin }^2}\theta }&{{{\cos }^2}\theta }&{1 + 4\sin 4\theta } \end{vmatrix} =0$
	If cot $(\alpha +\beta )=0,$then $\sin (\alpha +2\beta )=$
	If n is any integer, then the general solution of the equation $\cos x-\sin x=\frac{1}{\sqrt{2}}$ is
	The general solution of $\sin x-\cos x=\sqrt{2}$, for any integer n is
	If $12{{\cot }^{2}}\theta -31\,\text{cosec }\theta +\text{32}=\text{0}$, then the value of $\sin \theta $ is
	Period of $\|\sin 2x\|$ is
	Period of $\sin \theta \cos \theta $ is
	Period of $\frac{\sin \theta +\sin 2\theta }{\cos \theta +\cos 2\theta }$ is
	Period of $\cos (7x-5)$ is