In $\Delta ABC$,$(a - b)^2 \cos^2 \frac{C}{2} + (a + b)^2 \sin^2 \frac{C}{2} = $

The number of values of $\theta$ in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ such that $\theta \neq \frac{n \pi}{5}$ for $n=0, \pm 1, \pm 2$ and $\tan \theta = \cot 5 \theta$ as well as $\sin 2 \theta = \cos 4 \theta$ is

The number of roots of the equation $(81)^{\sin ^{2} x} + (81)^{\cos ^{2} x} = 30$ in the interval $[0, \pi]$ is equal to

In $\triangle ABC$,find the value of $r r_1 \cot \frac{A}{2} + r r_2 \cot \frac{B}{2} + r r_3 \cot \frac{C}{2}$.

If $\alpha, \beta, \gamma$ are angles of a triangle,then $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma - 2 \cos \alpha \cos \beta \cos \gamma$ is

The number of solutions to $\sin x = \frac{6}{x}$ with $0 \leq x \leq 12 \pi$ is