Trigonometric Functions and Equations

The function $f(x)=|\sin4x|+|\cos2x|$, is a periodic function with period :

The principal value of $\tan^{-1}\left(\cot\dfrac{43\pi}4\right)$ is :

The value of $\cot\left(\displaystyle\sum_{n=1}^{23}\cot^{-1}\left(1+\sum_{k=1}^n2k\right)\right)$ is

Let $f(x)=x\sin\pi x$, $x>0$. Then for all natural numbers $n$, $f'(x)$ vanishes at

Match List I with List II and select the correct answer using the code given below the lists:

Let $\theta,\varphi\in[0,2\pi]$ be such that
$$2\cos\theta(1-\sin\varphi)=\sin^2\theta\left(\tan\frac\theta2+\cot\frac\theta2\right)\cos\varphi-1,$$
$\tan(2\pi-\theta)>0$ and $-1<\sin\theta<-\dfrac{\sqrt3}2$. Then $\varphi$ cannot satisfy

Consider the statements:

P: There exists some $x\in\mathbb R$ such that $f(x)+2x=2(1+x^2)$

Q: There exists some $x\in\mathbb R$ such that $2f(x)+1=2x(1+x)$

Then

Let $f:(-1,1)\to\mathbb R$ be such that $f(\cos4\theta)=\dfrac2{2-\sec^2\theta}$ for $\theta\in\left(0,\dfrac\pi4\right)\cup\left(\dfrac\pi4,\dfrac\pi2\right)$. Then the value(s) of $f\left(\dfrac13\right)$ is (are)

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

The positive integer value of $n>3$ satisfying the equation
$$\frac1{\sin\left(\frac\pi n\right)}=\frac1{\sin\left(\frac{2\pi}n\right)}+\frac1{\sin\left(\frac{3\pi}n\right)}$$
is

Let $f(\theta)=\sin\left(\tan^{-1}\left(\dfrac{\sin\theta}{\sqrt{\cos2\theta}}\right)\right)$, where $-\dfrac\pi4<\theta<\dfrac\pi4$. Then the value of
$$\frac{d}{d(\tan\theta)}\left(f(\theta)\right)$$
is

Let $f(x)=x^2$ and $g(x)=\sin x$ for all $x\in\mathbb R$. Then the set of all $x$ satisfying $(f\circ g\circ g\circ f)(x)=(g\circ g\circ f)(x)$, where $(f\circ g)(x)=f(g(x))$, is

Match the statements given in Column I with the intervals/union of intervals given in Column II

The number of values of $\theta$ in the interval $\left(-\dfrac\pi2,\dfrac\pi2\right)$ such that $\theta\ne\dfrac{n\pi}5$ for $n=0,\pm1,\pm2$ and $\tan\theta=\cot5\theta$ as well as $\sin2\theta=\cos4\theta$ is

The maximum value of the expression $\dfrac1{\sin^2\theta+3\sin\theta\cos\theta+5\cos^2\theta}$ is

The number of all possible values of $\theta$, where $0<\theta<\pi$, for which the system of equations
$$(y+z)\cos3\theta=(xyz)\sin3\theta$$
$$x\sin3\theta=\frac{2\cos3\theta}y+\frac{2\sin3\theta}z$$
$$(xyz)\sin3\theta=(y+2z)\cos3\theta+y\sin3\theta$$
have a solution $(x_0,y_0,z_0)$ with $y_0z_0\ne0$, is

Two parallel chords of a circle of radius 2 are at a distance $\sqrt3+1$ apart. If the chords subtend at the center, angles of $\dfrac{\pi}{k}$ and $\dfrac{2\pi}{k}$, where $k>0$, then the value of $[k]$ is

[Note: $[k]$ denotes the largest integer less than or equal to $k$]

Match the statements in Column-I with the values in Column-II.

Let $z=\cos\theta+i\sin\theta$. Then the value of
$$\sum_{m=1}^{15}\operatorname{Im}\left(z^{2m-1}\right)$$
at $\theta=2°$ is

In a triangle $ABC$ with fixed base $BC$, the vertex $A$ moves such that
$$\cos B+\cos C=4\sin^2\frac A2.$$
If $a$, $b$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A$, $B$ and $C$, respectively, then