Trigonometric Equations

Number of solutions for $\tan^{-1}\sqrt{x(x+1)} + \sin^{-1}\sqrt{x^2 + x + 1} = \dfrac{\pi}{2}$ is:

The equation $\sin^4 x + \cos^4 x + \sin 2x + \alpha = 0$ is solvable for:

Solve: $ 2\sin^{2}\theta - 3\sin\theta - 2 = 0$

The solution of $\sin x +1 = \cos x $ such that $0\leq x\leq 2\pi$ is

Let $P = \{\theta : \sin\theta - \cos\theta = \sqrt{2}\cos\theta \}$ and $Q = \{\theta : \sin\theta + \cos\theta = \sqrt{2}\sin\theta \}$ be two sets. Then

The number of solutions of the equation sinx + sin5x = sin3x lying in the interval $[0, \pi]$ is

Let S be the set $\{a\in Z^+:a\leq100\}$.If the equation $[tan^2 x]-tan x - a = 0$ has real roots (where [ . ] is the greatest integer function), then the number of elements is S is