If $\frac{5 \sinh 2x}{7+6 \cosh 2x} = \frac{3}{2}$,then $3 \tanh^2 x + 20 \tanh x = $

If $a \sin^2 \theta + b \cos^2 \theta = c$,then $\tan^2 \theta$ is equal to

Let $\alpha = 3 + 4 + 8 + 9 + 13 + 14 + \dots$ up to $40$ terms. If $(\tan \beta)^{1020}$ is a root of the equation $x^2 + x - 2 = 0$, where $\beta \in (0, \frac{\pi}{2})$, then $\sin^2 \beta + 3 \cos^2 \beta$ is equal to:

If $\sec \theta = m$ and $\tan \theta = n$,then $\frac{1}{m} \left[ m + n + \frac{1}{m + n} \right] = $

If $\frac{\sqrt{2} \sin \alpha}{\sqrt{1+\cos 2 \alpha}}=\frac{1}{7}$ and $\sqrt{\frac{1-\cos 2 \beta}{2}}=\frac{1}{\sqrt{10}}$ where $\alpha, \beta \in (0, \frac{\pi}{2})$,then $\tan (\alpha+2 \beta)$ is equal to

If $2y \cos \theta = x \sin \theta$ and $2x \sec \theta - y \csc \theta = 3$,then $x^2 + 4y^2 = $