$\text{cosec } A - 2 \cot 2A \cos A = $

If $x \cos \theta = y \cos \left( \theta + \frac{2\pi}{3} \right) = z \cos \left( \theta + \frac{4\pi}{3} \right)$,then the value of $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ is equal to

If $\log_a b + \log_b c + \log_c a$ vanishes,where $a, b,$ and $c$ are positive real numbers different from unity,then the value of $(\log_a b)^3 + (\log_b c)^3 + (\log_c a)^3$ is

In triangle $ABC$,the value of $\sin 2A + \sin 2B + \sin 2C$ is equal to

In any triangle $ABC$,the expression ${\sin ^2}\frac{A}{2} + {\sin ^2}\frac{B}{2} + {\sin ^2}\frac{C}{2}$ is equal to:

The value of $(\sec \theta + \operatorname{cosec} \theta)$ when $\theta = 45^{\circ}$ is: (in $\sqrt{2}$)