Let $P = \left\{ {\theta :\sin \,\theta - \cos \,\theta = \sqrt 2 \,\cos \,\theta } \right\}$ and $Q = \left\{ {\theta :\sin \,\theta + \cos \,\theta = \sqrt {2\,} \sin \,\theta } \right\}$ be two sets. Then
$P \subset Q$ and $Q - P \ne \phi $
$Q \not\subset P$
$P = Q$
$P \not\subset Q$
The number of pairs $(x, y)$ satisfying the equations $\sin x + \sin y = \sin (x + y)$ and $|x| + |y| = 1$ is
The number of solutions of $\sin ^{7} x+\cos ^{7}=1, x \in[0,4 \pi]$ is equal to :
Let $\theta \in [0, 4\pi ]$ satisfy the equation $(sin\, \theta + 2) (sin\, \theta + 3) (sin\, \theta + 4) = 6$ . If the sum of all the values of $\theta $ is of the form $k\pi $, then the value of $k$ is
Let $f:[0,2] \rightarrow R$ be the function defined by
$f ( x )=(3-\sin (2 \pi x )) \sin \left(\pi x -\frac{\pi}{4}\right)-\sin \left(3 \pi x +\frac{\pi}{4}\right)$
If $\alpha, \beta \in[0,2]$ are such that $\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]$, then the value of $\beta-\alpha$ is. . . . . . . . .
The general solution of ${\sin ^2}\theta \sec \theta + \sqrt 3 \tan \theta = 0$ is