If $\tan \beta = \frac{\tan \alpha + \tan \gamma}{1 + \tan \alpha \tan \gamma}$,then $\frac{\sin 2 \alpha + \sin 2 \gamma}{1 + \sin 2 \alpha \sin 2 \gamma} = $

If $\tan \left(\frac{\pi}{9}\right), x, \tan \left(\frac{7 \pi}{18}\right)$ are in arithmetic progression and $\tan \left(\frac{\pi}{9}\right), y, \tan \left(\frac{5 \pi}{18}\right)$ are also in arithmetic progression,then $|x-2 y|$ is equal to:

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 $\tan A = \tan \alpha \coth x = \cot \beta \tanh x$,then $\tan (\alpha + \beta) =$

If $A$ lies in the third quadrant and $3 \tan A - 4 = 0$,then find the value of $5 \sin 2A + 3 \sin A + 4 \cos A$.

Evaluate the sum: $\sin^2(3^{\circ}) + \sin^2(6^{\circ}) + \sin^2(9^{\circ}) + \dots + \sin^2(84^{\circ}) + \sin^2(87^{\circ}) + \sin^2(90^{\circ})$