Trigonometric Equations

The solution set of the system of equations $X + Y = {{2\pi } \over 3},$ $cox\,x + cos\,y = {3 \over 2},$ where x and y are real, is _____.

The set of all $x$ in the interval $\left[ {0,\,\pi } \right]$ for which $2\,{\sin ^2}x - 3$ $\sin x + 1 \ge 0,$ is _____.

The sides of a triangle inscribed in a given circle subtend angles $\alpha $, $\beta $ and $\gamma $ at the centre. The minimum value of the arithmetic mean of $cos\left[ {\alpha + {\pi \over 2}} \right],\,\cos \left[ {\beta + {\pi \over 2}} \right]$ and $cos\left[ {\gamma + {\pi \over 2}} \right]$ is equal to _______.

Suppose ${\sin ^3}\,x\sin 3x = \sum\limits_{m = 0}^n {{C_m}\cos \,mx} $ is an identity in x, where C₀, C₁ ,....C_n are constants, and ${C_n} \ne 0$ , then the value of n is _____.

There exists a value of $\theta $ between 0 and $2\pi $ that satisfies the equation $\,\,{\sin ^4}\theta - 2{\sin ^2}\theta - 1 = 0.$

If $\tan \,A = \left( {1 - \cos B} \right)/\sin B,$ then $tan2A = tan\,B$.