Trigonometric Functions and Equations

$16\sin(20°)\sin(40°)\sin(80°)$ is equal to :

If the inverse trignometric functions take principal values, then
$$\cos^{-1}\left(\frac3{10}\cos\left(\tan^{-1}\left(\frac43\right)\right)+\frac25\sin\left(\tan^{-1}\left(\frac43\right)\right)\right)$$
is equal to :

If $\cos^{-1}\left(\dfrac y2\right)=\log_e\left(\dfrac x5\right)^5$, $|y|<2$, then :

The value of $\cos\left(\dfrac{2\pi}7\right)+\cos\left(\dfrac{4\pi}7\right)+\cos\left(\dfrac{6\pi}7\right)$ is equal to :

$\sin^{-1}\left(\sin\dfrac{2\pi}3\right)+\cos^{-1}\left(\cos\dfrac{7\pi}6\right)+\tan^{-1}\left(\tan\dfrac{3\pi}4\right)$ is equal to :

The value of $\cot\left(\displaystyle\sum_{n=1}^{50}\tan^{-1}\left(\frac1{1+n+n^2}\right)\right)$ is

$\alpha=\sin36°$ is a root of which of the following equation?

The value of $\displaystyle\lim_{n\to\infty}6\tan\left\{\sum_{r=1}^n\tan^{-1}\left(\frac1{r^2+3r+3}\right)\right\}$ is equal to :

If $\cot\alpha=1$ and $\sec\beta=-\dfrac53$, where $\pi<\alpha<\dfrac{3\pi}2$ and $\dfrac\pi2<\beta<\pi$, then the value of $\tan(\alpha+\beta)$ and the quadrant in which $\alpha+\beta$ lies, respectively are :

The distance between the two points A and A' which lie on $y=2$ such that both the line segments AB and A'B (where B is the point $(2,3)$) subtend angle $\dfrac\pi4$ at the origin, is equal to :

The domain of the function $\cos^{-1}\left(\dfrac{2\sin^{-1}\left(\frac1{4x^2-1}\right)}\pi\right)$ is :

The number of elements in the set $S=\left\{\theta\in[-4\pi,4\pi]:3\cos^22\theta+6\cos2\theta-10\cos^2\theta+5=0\right\}$ is ______.

The number of solutions of the equation $2\theta-\cos^2\theta+\sqrt2=0$ in $\mathbf R$ is equal to ______.

$50\tan\left(3\tan^{-1}\left(\dfrac12\right)+2\cos^{-1}\left(\dfrac1{\sqrt5}\right)\right)+4\sqrt2\tan\left(\dfrac12\tan^{-1}\left(2\sqrt2\right)\right)$ is equal to ______.

The number of solutions of the equation $\sin x=\cos^2x$ in the interval $(0,10)$ is ______.

Let m and M respectively be the minimum and the maximum values of $f(x)=\sin^{-1}2x+\sin2x+\cos^{-1}2x+\cos2x$, $x\in\left[0,\dfrac\pi8\right]$. Then $m+M$ is equal to :

Let $\alpha=\tan\left(\dfrac{5\pi}{16}\sin\left(2\cos^{-1}\left(\dfrac1{\sqrt5}\right)\right)\right)$ and $\beta=\cos\left(\sin^{-1}\left(\dfrac45\right)+\sec^{-1}\left(\dfrac53\right)\right)$ where the inverse trigonometric functions take principal values. Then, the equation whose roots are $\alpha$ and $\beta$ is :

Let $S_1=\left\{x\in[0,12\pi]:\sin^5x+\cos^5x=1\right\}$ and $S_2=\left\{x\in[0,8\pi]:\sin^7x+\cos^7x=1\right\}$. Then $n(S_1)-n(S_2)$ is equal to ______.

The number of $\theta\in(0,4\pi)$ for which the system of linear equations
$$3(\sin3\theta)x-y+z=2$$
$$3(\cos2\theta)x+4y+3z=3$$
$$6x+7y+7z=9$$
has no solution, is :

The number of solutions of $|\cos x|=\sin x$, such that $-4\pi\le x\le4\pi$ is :