If $\tan A = \frac{1}{2},\tan B = \frac{1}{3},$ then $\cos 2A = $

If $A + B + C = \pi ,$ then $\frac{{\cos A}}{{\sin B\sin C}} + \frac{{\cos B}}{{\sin C\sin A}} + \frac{{\cos C}}{{\sin A\sin B}} = $

If $\tan \theta = \frac{{\sin \alpha - \cos \alpha }}{{\sin \alpha + \cos \alpha }},$ then $\sin \alpha + \cos \alpha $ and $\sin \alpha - \cos \alpha $ must be equal to

For any $\theta \, \in \,\left( {\frac{\pi }{4},\frac{\pi }{2}} \right)$, the expression $3\,{\left( {\sin \,\theta  - \cos \,\theta } \right)^4} + 6{\left( {\sin \,\theta  + \cos \,\theta } \right)^2} + 4\,{\sin ^6}\,\theta $ equals

If $A + B + C = \frac{{3\pi }}{2},$ then $\cos 2A + \cos 2B + \cos 2C = $

If $0 <  x , y < \pi$ and $\cos x +\cos y-\cos ( x + y )=\frac{3}{2},$ then $\sin x+\cos y$ is equal to ...... .