If $\tan (\pi \cos \theta ) = \cot (\pi \sin \theta ),$ then the value of $\cos \left( {\theta - \frac{\pi }{4}} \right) =$
$\frac{1}{{2\sqrt 2 }}$
$\frac{1}{{\sqrt 2 }}$
$\frac{1}{{3\sqrt 2 }}$
$\frac{1}{{4\sqrt 2 }}$
$cos (\alpha \,-\,\beta ) = 1$ and $cos (\alpha +\beta ) = 1/e$ , where $\alpha , \beta \in [-\pi , \pi ]$ . Number of pairs of $(\alpha ,\beta )$ which satisfy both the equations is
If ${\sec ^2}\theta = \frac{4}{3}$, then the general value of $\theta $ is
If $0 \le x \le \pi $ and ${81^{{{\sin }^2}x}} + {81^{{{\cos }^2}x}} = 30$, then $x =$
The number of solutions of the equation $\sin (9 x)+\sin (3 x)=0$ in the closed interval $[0,2 \pi]$ is
The solution of $3\tan (A - {15^o}) = \tan (A + {15^o})$ is