JEE Main 2016 — Trigonometry

If $A>0, B>0$ and $A+B=\frac{\pi }{6}$, then the minimum positive value of $(\mathrm{tan}⁡A+\mathrm{tan}⁡B)$ is :

The number of $x \in [0, 2\pi ]$ for which $|\sqrt{2{\mathrm{sin}}^{4}x+18{\mathrm{cos}}^{2}x}- \sqrt{2{\mathrm{cos}}^{4}x+18{\mathrm{sin}}^{2}x}|=1$ is:

If $m$ and $M$ are the minimum and the maximum values of $4+\frac{1}{2}{\mathrm{sin}}^{2}⁡2x-2{\mathrm{cos}}^{4}⁡x, x \in R,$ then $M-m$ is equal to:

Let $P={\theta :\mathrm{sin}\theta -\mathrm{cos}\theta =\sqrt{2}\mathrm{cos}\theta }$ and $Q={\theta :\mathrm{sin}\theta +\mathrm{cos}\theta =\sqrt{2}\mathrm{sin}\theta },$ be two sets. Then

If $0 \leq x<2\pi ,$ then the number of real values of $x,$ which satisfy the equation $\mathrm{cos}x+\mathrm{cos}2x+\mathrm{cos}3x+\mathrm{cos}4x=0,$ is