All the pairs $(x, y)$ that satisfy the inequality ${2^{\sqrt {{{\sin }^2}{\kern 1pt} x - 2\sin {\kern 1pt} x + 5} }}.\frac{1}{{{4^{{{\sin }^2}\,y}}}} \leq 1$ also Satisfy the equation
$2\left| {\sin \,x} \right| = 3\sin \,y$
$\sin \,x = \left| {\sin \,y} \right|$
$2\,sin\, x = sin\, y$
$sin\, x = 2\, sin\, y$
The number of distinct solutions of $\sec \theta \,\, + \,\,\tan \theta \, = \,\sqrt 3 \,,\,0\,\, \leqslant \,\,\theta \,\, \leqslant \,\,2\pi$
If $0 \le x \le \pi $ and ${81^{{{\sin }^2}x}} + {81^{{{\cos }^2}x}} = 30$, then $x =$
For each positive real number $\lambda$. Let $A_\lambda$ be the set of all natural numbers $n$ such that $|\sin (\sqrt{n+1})-\sin (\sqrt{n})|<\lambda$. Let $A_\lambda^c$ be the complement of $A_\lambda$ in the set of all natural numbers. Then,
If $|cos\ x + sin\ x| + |cos\ x\ -\ sin\ x| = 2\ sin\ x$ ; $x \in [0,2 \pi ]$ , then maximum integral value of $x$ is
If $\tan 2\theta \tan \theta = 1$, then the general value of $\theta $ is