$$\cos \left( {\alpha - \beta } \right) = 1$$ and $$\,\cos \left( {\alpha + \beta } \right) = 1/e$$ where $$\alpha ,\,\beta \in \left[ { - \pi ,\pi } \right].$$
Paris of $$\alpha ,\,\beta $$ which satisfy both the equations is/are

Given both $$\theta $$ and $$\phi $$ are acute angles and $$\sin \,\theta = {1 \over 2},\,$$ $$\cos \,\phi = {1 \over 3},$$ then the value of $$\theta + \phi $$ belongs to

The number of integral values of $$k$$ for which the equation $$7\cos x + 5\sin x = 2k + 1$$ has a solution is

If $$\alpha + \beta = \pi /2$$ and $$\beta + \gamma = \alpha ,$$ then $$\tan \,\alpha \,$$ equals