Trigonometric equations practice · Mixed difficulty

If $5\,\,\cos \,\,2\theta + 2\,\,{\cos ^2}\,\,\frac{\theta }{2} + 1 = 0$, when $(0 < \theta < \pi )$, then the values of $\theta$ are

The set of all $x \in(-\pi, \pi)$ satisfying $|4 \sin x-1| < \sqrt{5}$ is given by

If $\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0$, then the number of distinct real roots of this equation in the interval $-\pi / 2 < x < \pi / 2$ is

If $\sin \,\,\left( {\pi \,\,\cot \,\,\theta } \right) = \cos \left( {\pi \,\,\tan \,\,\theta } \right)$, then

If $\exp ^{\left[\left(\sin ^2 x+\sin ^4 x+\sin ^6 x+\ldots\right) \ln 2\right]}$ satisfies the equation $y^2-9 y+8=0$, then the value of $\frac{\cos x}{\cos x+\sin x}, 0 < x < \frac{\pi}{2}$, is

Total number of solutions of $[\sin x]=\cos x$, where [ ] denotes the greatest integer function in $[0,3 \pi]$, is

From the identity $\sin 3 x=3 \sin x-4 \sin ^3 x$ it follows that if $x$ is real and $|x| < 1$, then

Total number of solutions of $\cos x \cos 2 x \cos 3 x=\frac{1}{4}$ in $[0, \pi]$ is

The number of roots of the equation $x + 2\,\,\tan \,\,x = \frac{\pi }{2}$ in the interval $\left[ {0,\,\,2\;\pi } \right]$, is

The general solution for $\theta$ if $\sin \left(2 \theta+\frac{\pi}{6}\right)+\cos \left(\theta+\frac{5 \pi}{6}\right)=2$ is

$\log \left(\sin 1^{\circ}\right) \log \left(\sin 2^{\circ}\right) \log \left(\sin 3^{\circ}\right) \ldots \log \left(\sin 179^{\circ}\right)$ is equal to

The number of values of $x \in \left[ {0,\,2\;\pi } \right]$ that satisfy $\cot \,\,x - \mathrm{cosec\;}\,\,x = 2\sin \,\,x$ is

The equation $e^{\sin x}-e^{-\sin x}-4=0$ has

If $-\pi \leq x \leq \pi,-\pi \leq y \leq \pi$ and $\cos x+\cos y=2$, then the value $\cos (x-y)$ is

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

The number of solutions of the equation ${32^{{{\tan }^2}x}} + {32^{{{\sec }^2}x}} = 81,\,\,0 \le x \le \frac{\pi }{4}$ is

The number of solutions for the equation $\sin \,\,2x + \cos \,\,4x = 2$ is

If $\tan \,\,\theta ,\,\,\cos \,\,\theta ,\,\,\frac{1}{6}\,\,\sin \,\,\theta$ are in G.P., then general value of $\theta$ is

If $\tan \,\,2x = \tan \,\,\frac{2}{x}$, then the value of $x$ is

$\sin \frac{2 \pi}{7}+\sin \frac{4 \pi}{7}+\sin \frac{8 \pi}{7}$ is equal to

If $\sin ^2 \theta+3 \cos \theta-2=0$, then $\cos ^3 \theta+\sec ^3 \theta=\ldots \ldots, \theta \neq(2 k+1) \frac{\pi}{2}, k \in Z$

A value of $\theta$ satisfying $\sin 5 \theta-\sin 3 \theta+\sin \theta=0$ such that $0 < \theta < \frac{\pi}{2}$ is

Total number of integral values of $n$ so that $\sin x(\sin x+\cos x)=n$ has atleast one solution, is

If $0 \leq x \leq \pi$ and $81^{\sin ^{2} x}+81^{\cos ^{2} x}=30$, then $x=$

If the expression $\frac{{\sin \,\,\frac{x}{2}\,\, + \,\,\cos \,\,\frac{x}{2} - \,\,i\,\,\tan \,\,x}}{{1 + 2\,i\,\,\sin \,\,\frac{x}{2}}}$ is real, then x is equal to

Consider the following statements.

I. Equal numbers may be added to (or subtracted from) both sides of an inequality.

II. When both sides are multiplied (or divided) by a negative number, then the inequality is reversed.

Choose the correct option.

If the equation ${\sin ^2}\,\,\mathrm{\theta } - \cos \,\,\mathrm{\theta } = \frac{1}{4}$, then the value of $\mathrm{\theta }$ lying in the interval $0\,\, \le \,\,\mathrm{\theta }\,\, \le \,\,2\mathrm{\pi }$ is

The general solution of the equations: $\sin \theta=-\frac{\sqrt{3}}{2}$ and $\tan \theta=\sqrt{3}$

If $a \cos 2 \theta+b \sin 2 \theta=c$ has $\alpha$ and $\beta$ as its solutions, then $\tan \alpha+\tan \beta$ and $\tan \alpha \tan \beta$ are, respectively

If $\mathrm{A}+\mathrm{B}+\mathrm{C}=\pi$, then $\cos \mathrm{A}+\cos \mathrm{B}-\cos \mathrm{C}=$