JEE Main 2015 — Trigonometry

If $f(x)=2{\mathrm{tan}}^{-1}x+{\mathrm{sin}}^{-1}(\frac{2x}{1+{x}^{2}}), x>1$, then $f(5)$ is equal to

If $\mathrm{cos}\alpha +\mathrm{cos}\beta =\frac{3}{2} \text{and} \mathrm{sin}\alpha +\mathrm{sin}\beta =\frac{1}{2}$and $\theta$ is the arithmetic mean of $\alpha &\beta ,$ then $\mathrm{sin}2\theta +\mathrm{cos}2\theta$ is equal to:

Let ${\mathrm{tan}}^{-1}y={\mathrm{tan}}^{-1}x+{\mathrm{tan}}^{-1}(\frac{2x}{1-{x}^{2}})$, where $|x|<\frac{1}{\sqrt{3}}$,Then a value of $y$ is

In a $\Delta ABC$, $\frac{a}{b}=2+\sqrt{3}$, and $\angle C=60^{\circ}.$ Then the ordered pair $(\angle A,\angle B)$ is equal to: