The numerical value of $\frac{\cos ^{2} 45^{\circ}}{\sin ^{2} 60^{\circ}}+\frac{\cos ^{2} 60^{\circ}}{\sin ^{2} 45^{\circ}}-\frac{\tan ^{2} 30^{\circ}}{\cot ^{2} 45^{\circ}}-\frac{\sin ^{2} 30^{\circ}}{\cot ^{2} 30^{\circ}}$ is

If ${x_1}, {x_2}, {x_3}, ..., {x_n}$ are in $A.P.$ whose common difference is $\alpha$,then the value of $\sin \alpha (\sec {x_1}\sec {x_2} + \sec {x_2}\sec {x_3} + ... + \sec {x_{n - 1}}\sec {x_n}) = $

If $\Delta ABC$ is right-angled at $B$,$AB = 6$ units,and $\angle C = 30^{\circ}$,then $AC$ is equal to (in units):

If $x = a \operatorname{cosec}^{n} \theta$ and $y = b \cot^{n} \theta$,then by eliminating $\theta$,we get:

If $\tan \left( \frac{\pi}{4} + \theta \right) + \tan \left( \frac{\pi}{4} - \theta \right) = \lambda \sec 2\theta$,then $\lambda$ =

Two towers $A$ and $B$ have heights $45 \, m$ and $15 \, m$ respectively. The angle of elevation from the bottom of tower $B$ to the top of tower $A$ is $60^{\circ}$. If the angle of elevation from the bottom of tower $A$ to the top of tower $B$ is $\theta$,then the value of $\sin \theta$ is: