Trigonometric Functions and Equations

If A > 0, B > 0 and $A+B=\dfrac\pi6$, then the minimum value of tanA + tanB is :

The number of distinct solutions of the equation
$$\frac54\cos^22x+\cos^4x+\sin^4x+\cos^6x+\sin^6x=2$$
in the interval $[0,2\pi]$ is

If $\alpha=3\sin^{-1}\left(\dfrac6{11}\right)$ and $\beta=3\cos^{-1}\left(\dfrac49\right)$, where the inverse trigonometric functions take only the principal values, then the correct option(s) is(are)

Let
$$\tan^{-1}y=\tan^{-1}x+\tan^{-1}\left(\frac{2x}{1-x^2}\right),$$
where $|x|<\dfrac1{\sqrt3}$. Then a value of $y$ is :

If $f(x)=2\tan^{-1}x+\sin^{-1}\left(\dfrac{2x}{1+x^2}\right)$, $x>1$, then $f(5)$ is equal to :

If $\cos\alpha+\cos\beta=\dfrac32$ and $\sin\alpha+\sin\beta=\dfrac12$ and θ is the arithmetic mean of α and β, then $\sin2\theta+\cos2\theta$ is equal to :

Let $f:[0,4\pi]\to[0,\pi]$ be defined by $f(x)=\cos^{-1}(\cos x)$. The number of points $x\in[0,4\pi]$ satisfying the equation
$$f(x)=\frac{10-x}{10}$$
is

For $x\in(0,\pi)$, the equation $\sin x+2\sin2x-\sin3x=3$ has

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

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

Let $f_k(x)=\dfrac1k\left(\sin^kx+\cos^kx\right)$ where $x\in\mathbf R$ and $k\ge1$. Then $f_4(x)-f_6(x)$ equals :

Let P be the relation defined on the set of all real numbers such that
$$P=\left\{(a,b):\sec^2a-\tan^2b=1\right\}.$$
Then P is :

$\displaystyle\int\frac{\sin^8x-\cos^8x}{1-2\sin^2x\cos^2x}\,dx$ is equal to :

The number of values of $\alpha$ in $[0,2\pi]$ for which $2\sin^3\alpha-7\sin^2\alpha+7\sin\alpha=2$, is :

If $\operatorname{cosec}\theta=\dfrac{p+q}{p-q}$ ($p\ne q\ne0$), then $\left|\cot\left(\dfrac\pi4+\dfrac\theta2\right)\right|$ is equal to :

Let $f$ be an odd function defined on the set of real numbers such that for $x\ge0$,
$$f(x)=3\sin x+4\cos x.$$
Then $f(x)$ at $x=-\dfrac{11\pi}6$ is equal to :

The integral $\displaystyle\int x\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)dx$ $(x>0)$ is equal to :

If $2\cos\theta+\sin\theta=1$ $\left(\theta\ne\dfrac\pi2\right)$, then $7\cos\theta+6\sin\theta$ is equal to :

Statement I : The equation $(\sin^{-1}x)^3+(\cos^{-1}x)^3-a\pi^3=0$ has a solution for all $a\ge\dfrac1{32}$.

Statement II : For any $x\in\mathbb R$, $\sin^{-1}x+\cos^{-1}x=\dfrac\pi2$ and $0\le\left(\sin^{-1}x-\dfrac\pi4\right)^2\le\dfrac{9\pi^2}{16}$.

If $f(\theta)=\begin{vmatrix}1&\cos\theta&1\\-\sin\theta&1&-\cos\theta\\-1&\sin\theta&1\end{vmatrix}$ and A and B are respectively the maximum and the minimum values of $f(\theta)$, then (A, B) is equal to :