Trigonometric Functions and Equations

Considering only the principal values of the inverse trigonometric functions, the value of
$$\cot^{-1}(\cot(-11))+10\sin\left(2\cos^{-1}\left(\frac1{\sqrt2}\right)\right)+10\sin\left(2\tan^{-1}(2)\right)$$
is

Let
$$\alpha=\left(1-2\cos\left(\frac\pi{11}\right)\right)\left(1-2\cos\left(\frac{3\pi}{11}\right)\right)\left(1-2\cos\left(\frac{9\pi}{11}\right)\right)\left(1-2\cos\left(\frac{27\pi}{11}\right)\right)\left(1-2\cos\left(\frac{81\pi}{11}\right)\right).$$
Then the value of $5-\alpha^2$ is ________.

Match each entry in List-I to the correct entry in List-II and choose the correct option.

The value of $n$ is ________.

If $\sin\left(\dfrac\pi{18}\right)\sin\left(\dfrac{5\pi}{18}\right)\sin\left(\dfrac{7\pi}{18}\right)=K$, then the value of $\sin\left(\dfrac{10K\pi}3\right)$ is :

Let $S=\{x\in[-\pi,\pi]:\sin x(\sin x+\cos x)=a,\ a\in\mathbb Z\}$. Then $n(S)$ is equal to :

Let $P=\{\theta\in[0,4\pi]:\tan^2\theta\ne1\}$ and $S=\{a\in\mathbb Z:2\left(\cos^8\theta-\sin^8\theta\right)\sec2\theta=a^2,\ \theta\in P\}$. Then $n(S)$ is :

Two adjacent sides of a parallelogram PQRS are given by $\overrightarrow{PQ}=\hat j+\hat k$ and $\overrightarrow{PS}=\hat i-\hat j$. If the side PS is rotated about the point P by an acute angle $\alpha$ in the plane of the parallelogram so that it becomes perpendicular to the side PQ, then $\sin^2\left(\dfrac{5\alpha}2\right)-\sin^2\left(\dfrac\alpha2\right)$ is equal to :

The value of $\displaystyle\int_0^{20\pi}\left(\sin^4x+\cos^4x\right)dx$ is equal to :

Let $[\cdot]$ denote the greatest integer function. If the domain of the function $f(x)=\cos^{-1}\left(\dfrac{4x+2[x]}3\right)$ is $[\alpha,\beta]$, then $12(\alpha+\beta)$ is equal to:

If $y=\tan^{-1}\left(\dfrac{3\cos x-4\sin x}{4\cos x+3\sin x}\right)+2\tan^{-1}\left(\dfrac x{1+\sqrt{1-x^2}}\right)$, then $\dfrac{dy}{dx}$ at $x=\dfrac{\sqrt3}2$ is equal to:

If $A=\dfrac{\sin3^\circ}{\cos9^\circ}+\dfrac{\sin9^\circ}{\cos27^\circ}+\dfrac{\sin27^\circ}{\cos81^\circ}$ and $B=\tan81^\circ-\tan3^\circ$, then $\dfrac BA$ is equal to ________.

Let $\vec{a_k}=(\tan\theta_k)\hat i+\hat j$ and $\vec{b_k}=\hat i-(\cot\theta_k)\hat j$, where $\theta_k=\dfrac{2^{k-1}\pi}{2^n+1}$, for some $n\in\mathbb N$, $n>5$. Then the value of $\dfrac{\sum_{k=1}^n\left|\vec{a_k}\right|^2}{\sum_{k=1}^n\left|\vec{b_k}\right|^2}$ is ________.

$\displaystyle\max_{0\le x\le\pi}\left(16\sin\left(\frac x2\right)\cos^3\left(\frac x2\right)\right)$ is equal to:

The integral $\displaystyle\int_0^1\cot^{-1}\left(1+x+x^2\right)dx$ is equal to:

Let $\tan A$, $\tan B$, where $A,B\in\left(-\dfrac\pi2,\dfrac\pi2\right)$, be the roots of the quadratic equation $x^2-2x-5=0$. Then $20\sin^2\left(\dfrac{A+B}2\right)$ is equal to:

The sum of all the integral values of $p$ such that the equation $3\sin^2x+12\cos x-3=p$, $x\in\mathbb R$, has at least one solution, is:

If $\dfrac\pi4+\displaystyle\sum_{p=1}^{11}\tan^{-1}\left(\frac{2^{p-1}}{1+2^{2p-1}}\right)=\alpha$, then $\tan\alpha$ is equal to ________.

Let $f(x)=\displaystyle\lim_{y\to0}\frac{(1-\cos(xy))\tan(xy)}{y^3}$. Then the number of solutions of the equation $f(x)=\sin x$, $x\in\mathbf R$ is :

If $S=\left\{\theta\in[-\pi,\pi]:\cos\theta\cos\dfrac{5\theta}2=\cos7\theta\cos\dfrac{7\theta}2\right\}$, then $n(S)$ is equal to ________.