Trigonometric Functions and Equations

The domain of the function $f(x)=\sin^{-1}\left(\dfrac{x^2-3x+2}{x^2+2x+7}\right)$ is :

Let $S=\left\{\theta\in[0,2\pi]-\left\{\dfrac\pi2,\dfrac{3\pi}2\right\}:\sin^{-1}(\sin\theta)+\cos^{-1}(\cos\theta)+\tan^{-1}(\tan\theta)=\dfrac{4\pi}5\right\}$. Then $\dfrac{30}\pi\displaystyle\sum_{\theta\in S}\theta$ is equal to _______.

For any positive integer $n$, let $S_n: (0, \infty) \to \mathbb R$ be defined by
$$S_n(x) = \sum_{k=1}^n \cot^{-1}\left(\frac{1 + k(k+1)x^2}{x}\right),$$
where for any $x \in \mathbb R$, $\cot^{-1}(x) \in (0, \pi)$ and $\tan^{-1}(x) \in \left(-\frac\pi2, \frac\pi2\right)$. Then which of the following statements is (are) TRUE?

The value of $\dfrac{16S_1}{\pi}$ is ____.

If $e^{\left(\cos^2x+\cos^4x+\cos^6x+\dots\infty\right)\log_e2}$ satisfies the equation $t^2-9t+8=0$, then the value of $\dfrac{2\sin x}{\sin x+\sqrt3\cos x}\left(0<x<\dfrac\pi2\right)$ is :

$\displaystyle\lim_{n\to\infty}\tan\left\{\sum_{r=1}^n\tan^{-1}\left(\frac1{1+r+r^2}\right)\right\}$ is equal to _______.

Let $f:[0,2]\to\mathbb R$ be the function defined by

$$f(x) = \left(3 - \sin(2\pi x)\right)\sin\left(\pi x - \frac\pi4\right) - \sin\left(3\pi x + \frac\pi4\right).$$

If $\alpha, \beta \in [0,2]$ are such that $\{x\in[0,2] : f(x)\ge 0\} = [\alpha,\beta]$, then the value of $\beta - \alpha$ is ______

The value of the limit

$$\lim_{x\to\frac\pi2}\frac{4\sqrt2(\sin3x + \sin x)}{\left(2\sin2x\sin\frac{3x}{2} + \cos\frac{5x}{2}\right) - \left(\sqrt2 + \sqrt2\cos2x + \cos\frac{3x}{2}\right)}$$

is ______

Let the function $f:(0,\pi)\to\mathbb R$ be defined by

$$f(\theta) = (\sin\theta + \cos\theta)^2 + (\sin\theta - \cos\theta)^4.$$

Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta\in\{\lambda_1\pi, \ldots, \lambda_r\pi\}$, where $0 < \lambda_1 < \cdots < \lambda_r < 1$. Then the value of $\lambda_1 + \cdots + \lambda_r$ is ______

Let

$$M = \begin{bmatrix}\sin^4\theta & -1-\sin^2\theta\\ 1+\cos^2\theta & \cos^4\theta\end{bmatrix} = \alpha I + \beta M^{-1},$$

where $\alpha = \alpha(\theta)$ and $\beta = \beta(\theta)$ are real numbers, and $I$ is the $2\times2$ identity matrix. If

$\alpha^*$ is the minimum of the set $\{\alpha(\theta) : \theta\in[0,2\pi)\}$ and

$\beta^*$ is the minimum of the set $\{\beta(\theta) : \theta\in[0,2\pi)\}$,

then the value of $\alpha^* + \beta^*$ is

For non-negative integers $n$, let

$$f(n) = \frac{\displaystyle\sum_{k=0}^n\sin\left(\frac{k+1}{n+2}\pi\right)\sin\left(\frac{k+2}{n+2}\pi\right)}{\displaystyle\sum_{k=0}^n\sin^2\left(\frac{k+1}{n+2}\pi\right)}$$

Assuming $\cos^{-1}x$ takes values in $[0,\pi]$, which of the following options is/are correct?

The value of

$$\sec^{-1}\left(\frac14\sum_{k=0}^{10}\sec\left(\frac{7\pi}{12} + \frac{k\pi}2\right)\sec\left(\frac{7\pi}{12} + \frac{(k+1)\pi}2\right)\right)$$

in the interval $\left[-\frac\pi4, \frac{3\pi}4\right]$ equals ______

Which of the following is the only CORRECT combination?

Which of the following is the only CORRECT combination?

The number of real solutions of the equation

$$\sin^{-1}\left(\sum_{i=1}^{\infty} x^{i+1} - x \sum_{i=1}^{\infty} \left(\frac{x}{2}\right)^i\right) = \frac{\pi}{2} - \cos^{-1}\left(\sum_{i=1}^{\infty} \left(-\frac{x}{2}\right)^i - \sum_{i=1}^{\infty} (-x)^i\right)$$

lying in the interval $\left(-\frac{1}{2}, \frac{1}{2}\right)$ is _____ .

(Here, the inverse trigonometric functions $\sin^{-1} x$ and $\cos^{-1} x$ assume values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $[0, \pi]$, respectively.)

Let $a, b, c$ be three non-zero real numbers such that the equation

$$\sqrt{3}\, a \cos x + 2b \sin x = c, \quad x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right],$$

has two distinct real roots $\alpha$ and $\beta$ with $\alpha + \beta = \dfrac{\pi}{3}$. Then, the value of $\dfrac{b}{a}$ is _____ .

For any positive integer $n$, define $f_n: (0, \infty) \to \mathbb{R}$ as

$$f_n(x) = \sum_{j=1}^{n} \tan^{-1}\left(\frac{1}{1 + (x+j)(x+j-1)}\right) \quad \text{for all } x \in (0, \infty).$$

(Here, the inverse trigonometric function $\tan^{-1} x$ assumes values in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.)

Then, which of the following statement(s) is (are) TRUE?

Let $E_1 = \left\{x \in \mathbb{R} : x \ne 1 \text{ and } \frac{x}{x-1} > 0\right\}$ and $E_2 = \left\{x \in E_1 : \sin^{-1}\left(\log_e\left(\frac{x}{x-1}\right)\right) \text{ is a real number}\right\}$.

(Here, the inverse trigonometric function $\sin^{-1} x$ assumes values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.)

Let $f: E_1 \to \mathbb{R}$ be the function defined by $f(x) = \log_e\left(\frac{x}{x-1}\right)$ and $g: E_2 \to \mathbb{R}$ be the function defined by $g(x) = \sin^{-1}\left(\log_e\left(\frac{x}{x-1}\right)\right)$.

The correct option is:

If $g(x) = \displaystyle\int_{\sin x}^{\sin(2x)} \sin^{-1}(t)\, dt$, then

Let $\alpha$ and $\beta$ be nonzero real numbers such that $2(\cos\beta - \cos\alpha) + \cos\alpha\cos\beta = 1$. Then which of the following is/are true?