Trigonometric equations practice · Mixed difficulty

If $A+B=\frac{\pi}{3}$ and $\cos A+\cos B=1$, then

The value of $x$ in $\left( {0,\;\,\frac{\pi }{2}} \right)$ satisfying the equation $\sin \,x\,\,cos\,\;x = \frac{1}{4}$ is

$\sin 47^{\circ}+\sin 61^{\circ}-\sin 11^{\circ}-\sin 25^{\circ}$ is equal to

If $\tan \,\,\mathrm{\theta } + \tan \,\,2\mathrm{\theta } + \sqrt 3 \,\,\tan \,\,\mathrm{\theta }\,\,\tan \,\,2\mathrm{\theta } = \sqrt 3$, then

Solution set of the inequality $|\tan x| < 1 / \sqrt{3}$ is

If $\sin \theta$ and $\cos \theta$ are the roots of the equation $a x^2-b x+c=0$, then $a, b$ and $c$ satisfy the relation

The values of $x$ between $\mathrm{\;0}$ and $2\,\pi$ which satisfy the equation $\sin \,\,x\,\sqrt {8\,\,{{\cos }^2}\,\,x} = 1$ are in AP. The common difference of the AP is

If $\tan \theta=\sqrt{n}$ for some non-square natural number $n$, then $\sec 2 \theta$ is

Consider the following statements:

I. The function $f(x)=$ greatest integer $\leq x, x \in R$ is a continuous function.

II. All trigonometric functions are continuous on $R$. Which of the statements given above is/are correct?

The value of $\left(1+\cos \frac{\pi}{8}\right)\left(1+\cos \frac{3 \pi}{8}\right)$ $\left(1+\cos \frac{5 \pi}{8}\right)\left(1+\cos \frac{7 \pi}{8}\right)$ is

If $0 \le x \le \pi$ and ${81^{{{\sin }^2}x}} + \,\,{81^{{{\cos }^2}x}} = 30$, then $x$ is equal to

The value of $\frac{\cot 54^{\circ}}{\tan 36^{\circ}}+\frac{\tan 20^{\circ}}{\cot 70^{\circ}}$ is

If ${\sec ^2}\theta = \sqrt 2 \left( {1 - {{\tan }^2}\theta } \right)$, then $\theta =$

General solution of equation $|\sin x|=\cos x$ is

The solution set of $\left( {5 + 4\cos \,\,\theta } \right)\,\,\left( {2\cos \,\,\theta + 1} \right) = 0$ in the interval $\left[ {0,\;\,2\pi } \right]$ is

If $\cot \left( {\mathrm{\alpha } + \mathrm{\beta }} \right) = 0,\mathrm{\;then}\,\,\sin \left( {\mathrm{\alpha } + 2\mathrm{\beta }} \right)$ is equal to

If $\tan 2\mathrm{\theta }\,\,\tan \mathrm{\theta } = 1$, then the general value of $\mathrm{\theta }$ is

The sum of all values of $x$ in $\left[ {0,\,\,2\pi } \right]$, for which $\sin x + \sin 2x + \sin 3x + \sin 4x = 0$, is equal to

The solution set of the equation $5 \cos 2 \theta+2 \cos ^2 \frac{\theta}{2}+1$ $=0$, in the interval $-\pi<\theta<\pi$, is

If $0 \le x < \frac{\pi }{2}$, then the number of values of $x$ for which $x - \sin 2x + \sin 3x = 0$, is

The set of real $x$ satisfying the inequality $\frac{5-2 x}{3} \leq \frac{x}{6}-5$ is $[a, \infty)$. The value of $a$ is

The equation $3 \sin ^2 x+10 \cos x-6=0$ is satisfied if

The general solution of $:\,\,{\sin ^2}\theta \,\,\sec \,\,\theta + \sqrt 3 \,\,\tan \,\,\theta = 0$ is

If $n$ is any integer, then the general solution of the equation $\cos \,\,\theta - \sin \,\,\theta = \frac{1}{{\sqrt 2 }}$ is

The number of values of $x$ in $\left[ {0,\,\,5\;\pi } \right]$ satisfying the equation $3\,\,\cos \,\,2x - 10\,\,\cos \,\,x + 7 = 0$ is

In $\triangle A B C$, if $\angle C=\frac{2 \pi}{3}$, then the value of $\cos ^2 A+\cos ^2 B-\cos A \cos B$ is

If $\cot \alpha$ equals the integral solution of the inequality $4 x^2-16 x+15<0$ and $\sin \beta$ equals to the slope of the bisector of the first quadrant, then $\sin (\alpha+\beta) \sin (\alpha-\beta)$ is equal to

The general solution of ${e^{ - 1/\sqrt 2 }}\,\,\left( {{e^{\sin x}} + \,\,{e^{\cos x}}} \right) = 2$ is

If $\cos ^{-1}(x / 2)+\cos ^{-1}(y / 3)=\theta$, then $9 x^2-12 x y \cos \theta$ $+4 y^2$ is equal to

If $\sin A+\cos A=\sqrt{2}$, then the value of $\cos ^2 A$ is