If $\alpha ,$ $\beta$ are different values of $x$ satisfying $a\cos x + b\sin x = c,$ then $\tan {\rm{ }}\left( {\frac{{\alpha + \beta }}{2}} \right) = $
$a + b$
$a - b$
$\frac{b}{a}$
$\frac{a}{b}$
The solution of the equation ${\cos ^2}x - 2\cos x = $ $4\sin x - \sin 2x,$ $\,(0 \le x \le \pi )$ is
The number of pairs $(x, y)$ satisfying the equations $\sin x + \sin y = \sin (x + y)$ and $|x| + |y| = 1$ is
Statement $-1:$ The number of common solutions of the trigonometric equations $2\,sin^2\,\theta - cos\,2\theta = 0$ and $2 \,cos^2\,\theta - 3\,sin\,\theta = 0$ in the interval $[0, 2\pi ]$ is two.
Statement $-2:$ The number of solutions of the equation, $2\,cos^2\,\theta - 3\,sin\,\theta = 0$ in the interval $[0, \pi ]$ is two.
If $\alpha ,\,\beta ,\,\gamma ,\,\delta $ are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity $k$ , then the value of $4\sin \frac{\alpha }{2} + 3\sin \frac{\beta }{2} + 2\sin \frac{\gamma }{2} + \sin \frac{\delta }{2}$ is equal to
The number of solution of the equation,$\sum\limits_{r = 1}^5 {\cos (r\,x)} $ $= 0$ lying in $(0, \pi)$ is :