Trigonometric Functions and Equations

Consider the following lists:

The correct option is:

Consider the ellipse
$$\frac{x^2}{4} + \frac{y^2}{3} = 1.$$
Let $H(\alpha, 0)$, $0 < \alpha < 2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent to the ellipse at the point $E$ intersects the positive $x$-axis at a point $G$. Suppose the straight line joining $F$ and the origin makes an angle $\phi$ with the positive $x$-axis.

The correct option is:

Let $\alpha$ and $\beta$ be real numbers such that $-\dfrac\pi4 < \beta < 0 < \alpha < \dfrac\pi4$. If $\sin(\alpha+\beta) = \dfrac13$ and $\cos(\alpha-\beta) = \dfrac23$, then the greatest integer less than or equal to
$$\left(\frac{\sin\alpha}{\cos\beta} + \frac{\cos\beta}{\sin\alpha} + \frac{\cos\alpha}{\sin\beta} + \frac{\sin\beta}{\cos\alpha}\right)^2$$
is _____.

Let $G$ be a circle of radius $R > 0$. Let $G_1, G_2, \ldots, G_n$ be $n$ circles of equal radius $r > 0$. Suppose each of the $n$ circles $G_1, G_2, \ldots, G_n$ touches the circle $G$ externally. Also, for $i = 1, 2, \ldots, n-1$, the circle $G_i$ touches $G_{i+1}$ externally, and $G_n$ touches $G_1$ externally. Then, which of the following statements is/are TRUE?

The set of all values of $k$ for which $\left(\tan^{-1}x\right)^3+\left(\cot^{-1}x\right)^3=k\pi^3$, $x\in\mathbb R$, is the interval :

The domain of the function $f(x)=\dfrac{\cos^{-1}\left(\dfrac{x^2-5x+6}{x^2-9}\right)}{\log_e\left(x^2-3x+2\right)}$ is :

Let $S=\left\{\theta\in[-\pi,\pi]-\left\{\pm\dfrac\pi2\right\}:\sin\theta\tan\theta+\tan\theta=\sin2\theta\right\}$. If $T=\displaystyle\sum_{\theta\in S}\cos2\theta$, then $T+n(S)$ is equal to :

Let $x*y=x^2+y^3$ and $(x*1)*1=x*(1*1)$.

Then a value of $2\sin^{-1}\left(\dfrac{x^4+x^2-2}{x^4+x^2+2}\right)$ is

The value of the integral
$$\int_{-\pi/2}^{\pi/2}\frac{dx}{\left(1+e^x\right)\left(\sin^6x+\cos^6x\right)}$$
is equal to

The number of solutions of the equation $\cos\left(x+\dfrac\pi3\right)\cos\left(\dfrac\pi3-x\right)=\dfrac14\cos^22x$, $x\in[-3\pi,3\pi]$ is:

If $y=\tan^{-1}\left(\sec x^3-\tan x^3\right)$, $\dfrac\pi2<x^3<\dfrac{3\pi}2$, then

The number of values of $x$ in the interval $\left(\dfrac\pi4,\dfrac{7\pi}4\right)$ for which $14\operatorname{cosec}^2x-2\sin^2x=21-4\cos^2x$ holds, is ______.

If $b_n=\displaystyle\int_0^{\frac\pi2}\frac{\cos^2nx}{\sin x}dx$, $n\in\mathbb N$, then

The value of $2\sin(12°)-\sin(72°)$ is :

The value of $\tan^{-1}\left(\dfrac{\cos\left(\frac{15\pi}4\right)-1}{\sin\left(\frac\pi4\right)}\right)$ is equal to :

$\displaystyle\lim_{x\to\frac1{\sqrt2}}\frac{\sin\left(\cos^{-1}x\right)-x}{1-\tan\left(\cos^{-1}x\right)}$ is equal to :

Let $f(x)=2\cos^{-1}x+4\cot^{-1}x-3x^2-2x+10$, $x\in[-1,1]$. If $[a,b]$ is the range of the function $f$, then $4a-b$ is equal to :

If $\sin^2(10°)\sin(20°)\sin(40°)\sin(50°)\sin(70°)=\alpha-\dfrac1{16}\sin(10°)$, then $16+\alpha^{-1}$ is equal to ______.

Let $S=(0,2\pi)-\left\{\dfrac\pi2,\dfrac{3\pi}4,\dfrac{3\pi}2,\dfrac{7\pi}4\right\}$. Let $y=y(x)$, $x\in S$, be the solution curve of the differential equation $\dfrac{dy}{dx}=\dfrac1{1+\sin2x}$, $y\left(\dfrac\pi4\right)=\dfrac12$. If the sum of abscissas of all the points of intersection of the curve $y=y(x)$ with the curve $y=\sqrt2\sin x$ is $\dfrac{k\pi}{12}$, then $k$ is equal to ______.

$\displaystyle\lim_{x\to0}\frac{\cos(\sin x)-\cos x}{x^4}$ is equal to :