Trigonometric equations practice · Mixed difficulty

If $\tan m \theta+\cot n \theta=0$, then the general value of $\theta$ is

The largest positive solution of $1 + {\sin ^4}x = {\cos ^2}3x$ in $\left[ { - \,5\pi /2,\;\,5\pi /2} \right]$ is

If $\tan \,\,\left( {\cot \,\,x} \right) = \cot \,\,\left( {\tan \,\,x} \right)$, then $\sin \,\,2x$ is equal to

Number of solutions of the equation $\sin \,\,2\theta + 2 = 4\,\,\sin \,\,\theta + \cos \,\,\theta$ lying in the interval $\left[ {\pi ,\,\,5\;\pi } \right]$, is

Consider the following statements.

I. Inequalities involving the symbol $\geq$ or $\leq$ are called slack inequalities.

II. Inequalities which do not involve variables are called numerical inequalities.

Choose the correct option.

If $x=\sin ^2 \theta \cos \theta$ and $y=\cos ^2 \theta \sin \theta$, then which of the following is true ? $0 < \theta < \frac{\pi}{2}, x > y > 0$

The value of $3\left[\sin ^4\left(\frac{3 \pi}{2}-\alpha\right)+\sin ^4(3 \pi+\alpha)\right]$ $-2\left[\sin ^6\left(\frac{\pi}{2}+\alpha\right)+\sin ^6(5 \pi-\alpha)\right]$ is

The general value of $\mathrm{\theta }$ satisfying ${\sin ^2}\,\,\mathrm{\theta } + \sin \,\,\mathrm{\theta } = 2$ is

The solutions of the equation $4\,\,{\cos ^2}\,\,x + 6\,\,{\sin ^2}\,\,x = 5$ are

Number of integral value(s) of $m$ for which the equation $\sin x-\sqrt{3} \cos x=\frac{4 m-6}{4-m}$ has solutions, $x \in[0,2 \pi]$, is___.

$\frac{\sin 3 \theta}{2 \cos 2 \theta+1}=\frac{1}{2}$ if -

If the expression $n \sin ^2 \theta+2 n \cos (\theta+\alpha) \sin \alpha \sin \theta$ $+\cos 2(\alpha+\theta)$ is independent of $\theta$, then the value of $n$ is

The solution set of $\left( {2\cos x - 1} \right)\,\,\left( {3 + 2\,\cos x} \right) = 0$ in the interval $0 \le x \le 2\;\pi$, is

The general solution of the equation $\cos \,\,x\,\;\mathrm{cos}\,\;6x = - 1$ is

If $\sin x+\sqrt{3} \cos x=\sqrt{2}$, then $x$ is equal to

The number of solutions of the equation $\sin x = \cos \,\,3x\,\,\;in\;\,\left[ {0,\,\,\pi } \right]$, is

The equation $\sin x \cos x=2$ has

If $\frac{\sin x}{\sin y}=\frac{1}{2}, \frac{\cos x}{\cos y}=\frac{3}{2}$, where $x, y \in\left(0, \frac{\pi}{2}\right)$, then the value of $\tan (x+y)$ is

The equation $a \sin x+b \cos x=c$, where $|c|>\sqrt{a^2+b^2}$ has

Let $\Delta=\left|\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right|$, then $\Delta$ lies in the interval.

The most general solution of

${2^{1 + \,\left| {\,\cos x\,} \right| + \,{{\cos }^2}x + \,\left| {\,{{\cos }^3}x\,} \right| + \ldots \infty }} = 4$ is given by

Let $f(\theta)=3\left(\sin ^{4}\left(\frac{3 \pi}{2}-\theta\right)+\sin ^{4}(3 \pi+\theta)\right)-2\left(1-\sin ^{2} 2 \theta\right)$ and $S=\left\{\theta \in[0, \pi]: f^{\prime}(\theta)=-\frac{\sqrt{3}}{2}\right\}$. If $4 \beta=\sum_{\theta \in S} \theta$, then $f(\beta)$ is equal to

If $r > 0,-\pi \leq \theta \leq \pi$ and $r, \theta$ satisfy $r \sin \theta=3$ and $r=4(1+\sin \theta)$, then the number of possible solutions of the pair $(r, \theta)$ is

Statement-1: The number of common solutions of the trigonometric equations $2 \sin ^{2} \theta-\cos 2 \theta=0$ and $2 \cos ^{2} \theta-3 \sin \theta=0$ in the interval $[0,2 \pi]$ is two.

Statement-2: The number of solutions of the equation, 2 $\cos ^{2} \theta-3 \sin \theta=0$ in the interval $[0, \pi]$ is two.

If $\sin \theta+\operatorname{cosec} \theta=2$, then $\sin ^n \theta+\operatorname{cosec}^n \theta$ is equal to

If $0 < \beta < \alpha < \frac{\pi}{4}, \cos (\alpha+\beta)=\frac{3}{5}$ and $\cos (\alpha-\beta)=\frac{4}{5}$, then $\sin 2 \alpha$ is equal to

If $1+\sin \theta+\sin ^2 \theta+\ldots \ldots \infty=4+2 \sqrt{3}, 0 < \theta < \pi, \theta \neq \frac{\pi}{2}$, then

The numerical value of $\sqrt{3} \operatorname{cosec} 20^{\circ}-\sec 20^{\circ}$ is

The number of solutions of $\sin 3 x=\cos 2 x$, in the interval $\left(\frac{\pi}{2}, \pi\right)$ is

If $x$ and $y$ are acute angles such that $\cos x+\cos y=\frac{3}{2}$ and $\sin x+\sin y=\frac{3}{4}$, then $\sin (x+y)$ equals