Trigonometric Functions and Equations

$2\sin\left(\dfrac\pi{22}\right)\sin\left(\dfrac{3\pi}{22}\right)\sin\left(\dfrac{5\pi}{22}\right)\sin\left(\dfrac{7\pi}{22}\right)\sin\left(\dfrac{9\pi}{22}\right)$ is equal to :

Let $x=\sin\left(2\tan^{-1}\alpha\right)$ and $y=\sin\left(\dfrac12\tan^{-1}\dfrac43\right)$. If $S=\left\{\alpha\in\mathbb R:y^2=1-x\right\}$, then $\displaystyle\sum_{\alpha\in S}16\alpha^3$ is equal to ______.

Let $S=\left\{\theta\in[0,2\pi]:8^{2\sin^2\theta}+8^{2\cos^2\theta}=16\right\}$. Then $n(S)+\displaystyle\sum_{\theta\in S}\left(\sec\left(\frac\pi4+2\theta\right)\operatorname{cosec}\left(\frac\pi4+2\theta\right)\right)$ is equal to :

$\tan\left(2\tan^{-1}\dfrac15+\sec^{-1}\dfrac{\sqrt5}2+2\tan^{-1}\dfrac18\right)$ is equal to :

If the maximum value of $a$, for which the function $f_a(x)=\tan^{-1}2x-3ax+7$ is non-decreasing in $\left(-\dfrac\pi6,\dfrac\pi6\right)$, is $\bar a$, then $f_{\bar a}\left(\dfrac\pi8\right)$ is equal to

If $0<x<\dfrac1{\sqrt2}$ and $\dfrac{\sin^{-1}x}{\alpha}=\dfrac{\cos^{-1}x}{\beta}$, then a value of $\sin\left(\dfrac{2\pi\alpha}{\alpha+\beta}\right)$ is

If the sum of solutions of the system of equations $2\sin^2\theta-\cos2\theta=0$ and $2\cos^2\theta+3\sin\theta=0$ in the interval $[0,2\pi]$ is $k\pi$, then $k$ is equal to ______.

For $k\in\mathbf R$, let the solutions of the equation $\cos\left(\sin^{-1}\left(x\cot\left(\tan^{-1}\left(\cos\left(\sin^{-1}x\right)\right)\right)\right)\right)=k$, $0<|x|<\dfrac1{\sqrt2}$ be $\alpha$ and $\beta$, where the inverse trigonometric functions take only principal values. If the solutions of the equation $x^2-bx-5=0$ are $\dfrac1{\alpha^2}+\dfrac1{\beta^2}$ and $\dfrac\alpha\beta$, then $\dfrac b{k^2}$ is equal to ______.

The domain of the function $f(x)=\sin^{-1}\left[2x^2-3\right]+\log_2\left(\log_{\frac12}\left(x^2-5x+5\right)\right)$, where $[t]$ is the greatest integer function, is :

Let S be the set of all $(\alpha,\beta)$, $\pi<\alpha,\beta<2\pi$, for which the complex number $\dfrac{1-i\sin\alpha}{1+2i\sin\alpha}$ is purely imaginary and $\dfrac{1+i\cos\beta}{1-2i\cos\beta}$ is purely real. Let $Z_{\alpha\beta}=\sin2\alpha+i\cos2\beta$, $(\alpha,\beta)\in S$. Then $\displaystyle\sum_{(\alpha,\beta)\in S}\left(iZ_{\alpha\beta}+\frac1{i\bar Z_{\alpha\beta}}\right)$ is equal to :

Let $S=\left\{\theta\in\left(0,\dfrac\pi2\right):\displaystyle\sum_{m=1}^9\sec\left(\theta+(m-1)\frac\pi6\right)\sec\left(\theta+\frac{m\pi}6\right)=-\frac8{\sqrt3}\right\}$. Then

Considering only the principal values of the inverse trigonometric functions, the domain of the function $f(x)=\cos^{-1}\left(\dfrac{x^2-4x+2}{x^2+3}\right)$ is :

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation $\cos^{-1}(x)-2\sin^{-1}(x)=\cos^{-1}(2x)$ is equal to :

The sum of the absolute maximum and absolute minimum values of the function $f(x)=\tan^{-1}(\sin x-\cos x)$ in the interval $[0,\pi]$ is :

Let $S=\left[-\pi,\dfrac\pi2\right)-\left\{-\dfrac\pi2,-\dfrac\pi4,-\dfrac{3\pi}4,\dfrac\pi4\right\}$. Then the number of elements in the set $A=\left\{\theta\in S:\tan\theta\left(1+\sqrt5\tan(2\theta)\right)=\sqrt5-\tan(2\theta)\right\}$ is ______.

The value of the integral $\displaystyle\int_0^{\pi/2}60\frac{\sin(6x)}{\sin x}dx$ is equal to ______.

Let S = {θ ∈ (0, 2π) : $7\cos^2\theta-3\sin^2\theta-2\cos^22\theta=2$}. Then, the sum of roots of all the equations $x^2-2(\tan^2\theta+\cot^2\theta)x+6\sin^2\theta=0$, θ ∈ S, is _______.

For $I(x)=\displaystyle\int\frac{\sec^2x-2022}{\sin^{2022}x}\,dx$, if $I\left(\dfrac\pi4\right)=2^{1011}$, then

Let $m_1$, $m_2$ be the slopes of two adjacent sides of a square of side a such that $a^2+11a+3(m_1^2+m_2^2)=220$. If one vertex of the square is $(10(\cos\alpha-\sin\alpha),\ 10(\sin\alpha+\cos\alpha))$, where $\alpha\in\left(0,\dfrac\pi2\right)$ and the equation of one diagonal is $(\cos\alpha-\sin\alpha)x+(\sin\alpha+\cos\alpha)y=10$, then $72(\sin^4\alpha+\cos^4\alpha)+a^2-3a+13$ is equal to :

The number of elements in the set $S=\left\{x\in\mathbb R:2\cos\left(\dfrac{x^2+x}6\right)=4^x+4^{-x}\right\}$ is :