Trigonometric Functions and Equations

$|\overrightarrow{OX} \times \overrightarrow{OY}| =$

If the triangle $PQR$ varies, then the minimum value of

$$\cos(P + Q) + \cos(Q + R) + \cos(R + P)$$

is

If for $x\in\left(0,\dfrac14\right)$, the derivative of $\tan^{-1}\left(\dfrac{6x\sqrt x}{1-9x^3}\right)$ is $\sqrt x\cdot g(x)$, then $g(x)$ equals :

If $5(\tan^2x-\cos^2x)=2\cos2x+9$, then the value of $\cos4x$ is :

The integral $\displaystyle\int\sqrt{1+2\cot x(\operatorname{cosec}x+\cot x)}\,dx$ $\left(0<x<\dfrac\pi2\right)$ is equal to :

(where C is a constant of integration)

If two parallel chords of a circle, having diameter 4 units, lie on the opposite sides of the centre and subtend angles $\cos^{-1}\left(\dfrac17\right)$ and $\sec^{-1}(7)$ at the centre respectively, then the distance between these chords, is :

If $S=\left\{x\in[0,2\pi]:\begin{vmatrix}0&\cos x&-\sin x\\\sin x&0&\cos x\\\cos x&\sin x&0\end{vmatrix}=0\right\}$, then $\displaystyle\sum_{x\in S}\tan\left(\frac\pi3+x\right)$ is equal to :

The value of $\tan^{-1}\left[\dfrac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right]$, $|x|<\dfrac12$, $x\ne0$, is equal to :

A value of $x$ satisfying the equation $\sin[\cot^{-1}(1+x)]=\cos[\tan^{-1}x]$, is :

Let $-\dfrac{\pi}{6}<\theta<-\dfrac{\pi}{12}$. Suppose $\alpha_1$ and $\beta_1$ are the roots of the equation $x^2-2x\sec\theta+1=0$ and $\alpha_2$ and $\beta_2$ are the roots of the equation $x^2+2x\tan\theta-1=0$. If $\alpha_1>\beta_1$ and $\alpha_2>\beta_2$, then $\alpha_1+\beta_2$ equals

Let $S=\left\{x\in(-\pi,\pi):x\ne0,\pm\dfrac{\pi}{2}\right\}$. The sum of all distinct solutions of the equation $\sqrt3\sec x+\operatorname{cosec}x+2(\tan x-\cot x)=0$ in the set $S$ is equal to

The value of $\displaystyle\sum_{k=1}^{13}\frac{1}{\sin\left(\frac{\pi}{4}+\frac{(k-1)\pi}{6}\right)\sin\left(\frac{\pi}{4}+\frac{k\pi}{6}\right)}$ is equal to

Consider $f(x)=\tan^{-1}\left(\sqrt{\dfrac{1+\sin x}{1-\sin x}}\right)$, $x\in\left(0,\dfrac\pi2\right)$. A normal to y = f(x) at $x=\dfrac\pi6$ also passes through the point :

If $0\le x<2\pi$, then the number of real values of $x$, which satisfy the equation $\cos x+\cos2x+\cos3x+\cos4x=0$, is :

The number of distinct real roots of the equation, $\begin{vmatrix}\cos x&\sin x&\sin x\\\sin x&\cos x&\sin x\\\sin x&\sin x&\cos x\end{vmatrix}=0$ in the interval $\left[-\dfrac\pi4,\dfrac\pi4\right]$ is :

If $2\displaystyle\int_0^1\tan^{-1}x\,dx=\int_0^1\cot^{-1}(1-x+x^2)\,dx$, then $\displaystyle\int_0^1\tan^{-1}(1-x+x^2)\,dx$ is equal to :

The number of $x\in[0,2\pi]$ for which $\left|\sqrt{2\sin^4x+18\cos^2x}-\sqrt{2\cos^4x+18\sin^2x}\right|=1$ is :

If m and M are the minimum and the maximum values of $4+\dfrac12\sin^22x-2\cos^4x$, $x\in\mathbb R$, then M − m is equal to :

Let $P=\{\theta:\sin\theta-\cos\theta=\sqrt2\cos\theta\}$ and $Q=\{\theta:\sin\theta+\cos\theta=\sqrt2\sin\theta\}$ be two sets. Then :

Let $f(x)=\sin^4x+\cos^4x$. Then f is an increasing function in the interval :