JEE Main 2017 — Trigonometry

The lengths of two adjacent sides of a cyclic quadrilateral are $2$ units and $5$ units and the angle between them is ${60}^{o}$. If the area of the quadrilateral is $4\sqrt{3}$ sq. units, then the perimeter of the quadrilateral is

A value of $x$ satisfying the equation $\mathrm{sin}⁡[{\mathrm{cot}}^{-1}⁡(1+x)]=\mathrm{cos}⁡[{\mathrm{tan}}^{-1}⁡x],$ is:

The value of ${\mathrm{tan}}^{-1}⁡[\frac{\sqrt{1+{x}^{2}}+ \sqrt{1-{x}^{2}}}{\sqrt{1+{x}^{2}}- \sqrt{1-{x}^{2}}}],$ $|x|<\frac{1}{2}, x\neq 0,$ is equal to:

If $5({\mathrm{tan}}^{2}⁡x-{\mathrm{cos}}^{2}⁡x)=2\mathrm{cos}⁡ 2x+9,$ then the value of $\mathrm{cos}⁡4x$ is