$\tan 5x \tan 3x \tan 2x = $

The value of $\cos ^{2} 75^{\circ}+\cos ^{2} 45^{\circ}+\cos ^{2} 15^{\circ}-\cos ^{2} 30^{\circ}-\cos ^{2} 60^{\circ}$ is

Suppose $\theta_1$ and $\theta_2$ are such that $(\theta_1-\theta_2)$ lies in the $3^{\text{rd}}$ or $4^{\text{th}}$ quadrant. If $\sin \theta_1+\sin \theta_2=-\frac{21}{65}$ and $\cos \theta_1+\cos \theta_2=-\frac{27}{65}$,then $\cos \left(\frac{\theta_1-\theta_2}{2}\right)=$

If $\tan A = \frac{1}{\sqrt{x(x^2+x+1)}}, \tan B = \frac{\sqrt{x}}{\sqrt{x^2+x+1}}$ and $\tan C = (x^{-3}+x^{-2}+x^{-1})^{\frac{1}{2}}$,where $0 < A, B, C < \frac{\pi}{2}$,then $A+B$ is equal to:

If $\sin A = -\frac{24}{25}$,$\cos B = \frac{15}{17}$,$A$ does not belong to the $4^{\text{th}}$ quadrant,and $B$ does not belong to the $1^{\text{st}}$ quadrant,then $(A+B)$ lies in which quadrant?

If $\cos (A - B) = \frac{3}{5}$ and $\tan A \tan B = 2$,then