Trigonometric Ratio and Identites

If cos($$\alpha$$ + $$\beta$$) = 3/5 ,sin ( $$\alpha$$ - $$\beta$$) = 5/13 and 0 < $$\alpha , \beta$$ < $$\pi \over 4$$, then tan(2$$\alpha$$) is equal to :

If 0 < x, y < $$\pi$$ and cosx + cosy $$-$$ cos(x + y) = $${3 \over 2}$$, then sinx + cosy is equal to :

If $$\tan \left( {{\pi \over 9}} \right),x,\tan \left( {{{7\pi } \over {18}}} \right)$$ are in arithmetic progression and $$\tan \left( {{\pi \over 9}} \right),y,\tan \left( {{{5\pi } \over {18}}} \right)$$ are also in arithmetic progression, then $$|x - 2y|$$ is equal to :

If cot$$\alpha$$ = 1 and sec$$\beta$$ = $$- {5 \over 3}$$, where $$\pi < \alpha < {{3\pi } \over 2}$$ and $${\pi \over 2} < \beta < \pi$$, then the value of $$\tan (\alpha + \beta )$$ and the quadrant in which $$\alpha$$ + $$\beta$$ lies, respectively are :

If $\tan \mathrm{A}=\frac{1}{\sqrt{x\left(x^2+x+1\right)}}, \tan \mathrm{B}=\frac{\sqrt{x}}{\sqrt{x^2+x+1}}$ and $\tan \mathrm{C}=\left(x^{-3}+x^{-2}+x^{-1}\right)^{1 / 2}, 0<\mathrm{A}, \mathrm{B}, \mathrm{C}<\frac{\pi}{2}$, then $\mathrm{A}+\mathrm{B}$ is equal to :

For $$\alpha, \beta \in(0, \pi / 2)$$, let $$3 \sin (\alpha+\beta)=2 \sin (\alpha-\beta)$$ and a real number $$k$$ be such that $$\tan \alpha=k \tan \beta$$. Then, the value of $$k$$ is equal to

The number of solutions, of the equation $$e^{\sin x}-2 e^{-\sin x}=2$$, is :

If $$0 < x < \pi$$ and $$\cos x + \sin x = {1 \over 2},$$ then $$\tan x$$ is :

Let $$f_k\left( x \right) = {1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)$$ where $$x \in R$$ and $$k \ge \,1.$$ Then $${f_4}\left( x \right) - {f_6}\left( x \right)\,\,$$ equals :

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