Questions in trigonometry

Select	Question
	$\cot (45{}^\circ +\theta )\cot (45{}^\circ -\theta )=$
	$\tan A+\cot (180{}^\circ +A)+\cot (90{}^\circ +A)+\cot (360{}^\circ -A)$
	$\tan \theta \sin \left( \frac{\pi }{2}+\theta \right)\cos \left( \frac{\pi }{2}-\theta \right)=$
	If angle $\theta $ be divided into two parts such that the tangent of one part is $k$ times the tangent of the other and $\varphi $ is their difference, then $\sin \theta =$
	If $x=y\cos \frac{2\pi }{3}=z\cos \frac{4\pi }{3}$, then $xy+yz+zx=$
	Given that $\pi <\alpha <\frac{3\pi }{2},$ then the expression $\sqrt{(4{{\sin }^{4}}\alpha +{{\sin }^{2}}2\alpha )}+4{{\cos }^{2}}\left( \frac{\pi }{4}-\frac{\alpha }{2} \right)$ is equal to
	${{\sin }^{2}}\frac{\pi }{8}+{{\sin }^{2}}\frac{3\pi }{8}+{{\sin }^{2}}\frac{5\pi }{8}+{{\sin }^{2}}\frac{7\pi }{8}=$
	$(\sec A+\tan A-1)(\sec A-\tan A+1)-2\tan A=$
	The value of $\tan (-945{}^\circ )$ is
	If $\tan A=\frac{1}{2},\tan B=\frac{1}{3},$then $\cos 2A=$