If $2y\,\cos \theta = x\sin \,\theta {\rm{ and }}2x\sec \theta - y\,{\rm{cosec}}\,\theta = 3,$ then ${x^2} + 4{y^2} = $
$4$
$-4$
$± 4$
None of these
If ${\rm{cosec }}A + \cot A = \frac{{11}}{2},$ then $\tan A = $
If $0 < x < \pi $ and $\cos x + \sin x = \frac{1}{2}$,then $tan \,x$ is
If $a\,{\cos ^3}\alpha + 3a\,\cos \alpha \,{\sin ^2}\alpha = m$ and $a\,{\sin ^3}\alpha + 3a\,{\cos ^2}\alpha \sin \alpha = n,$ then ${(m + n)^{2/3}} + {(m - n)^{2/3}}$ is equal to
If $x\sin 45^\circ {\cos ^2}60^\circ = \frac{{{{\tan }^2}60^\circ {\rm{cosec}}30^\circ }}{{\sec 45^\circ {{\cot }^2}30^\circ }},$ then $x = $
Prove that:
$\sin ^{2} \frac{\pi}{6}+\cos ^{2} \frac{\pi}{3}-\tan ^{2} \frac{\pi}{4}=-\frac{1}{2}$