The equation $2{\cos }^2\left(\frac{x}{2}\right){\sin }^{2}x=x^2+x^{-2}, x\le \frac{\pi }{9}$ has

Important Questions on Trigonometric Ratios and Identities

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

If  $f\left(x\right)=\sqrt{\left[\right(\sin 2x+2\left)\right(\cos 2x+2\left)\right]}$ for $x$ is a real number, then which of the following values lies in the domain of $f\left(x\right)$ but NOT in the range of $f\left(x\right) ?$

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The number of solutions of the equation $\left|\cot x\right|=\cot x+\frac{1}{\sin x},0\le x\le 2\pi$

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The equation $\sin \left(e^x\right)=5^x+5^{-x}$ has exactly how many solutions?

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The value of $\tan 1°\tan 2°\tan 3°\dots \tan 89°$ is

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

If $A=\left\{x\in [0,2\pi ]/\tan x-{\tan }^{2}x>0\right\}$ and $B=\left\{x\in [0,2\pi ]/\left|\sin x\right|<\frac{1}{2}\right\},$ then $\mathrm{A}\cap \mathrm{B}=$

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The value of $\cos 1200°+\tan 1485°$ is

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

If $x=\cos 1°\cos 2°\cos 3°\dots \cos 89°$ and $y=\cos 2°\cos 6°\cos 10°\dots \cos 86°,$ then what is the integer nearest to $\frac{2}{7}{\log }_2\left(\frac{y}{x}\right)?$ 

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

Let $\theta$ be an angle in the standard position such that the point $(-5,12)$ lies on its terminal side, then

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The graphs of the polynomial $x^2-1$ and $\cos x$ intersect

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

If $\cos \theta =-\frac{\sqrt{3}}{2}$ and $\sin \alpha =-\frac{3}{5}$, where $\theta$ does not lie in the third quadrant, then $\frac{2\tan \alpha +\sqrt{3}\tan \theta }{{\cot }^2\theta +\cos \alpha }$ is equal to

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The range of the function $f\left(x\right)={\log }_{\sqrt{5}}\left(3+\cos \left(\frac{3\pi }{4}+x\right)+\cos \left(\frac{\pi }{4}+x\right)+\cos \left(\frac{\pi }{4}-x\right)-\cos \left(\frac{3\pi }{4}-x\right)\right)$ is :

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

$\tan 1°\times \tan 2°\times \tan 3°\times \cdots \cdots \cdots \cdots \times \tan 89°=$

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The value of $2\sin \left(\frac{\mathrm{\pi }}{8}\right)\sin \left(\frac{2\mathrm{\pi }}{8}\right)\sin \left(\frac{3\mathrm{\pi }}{8}\right)\sin \left(\frac{5\mathrm{\pi }}{8}\right)\sin \left(\frac{6\mathrm{\pi }}{8}\right)\sin \left(\frac{7\mathrm{\pi }}{8}\right)$ is :

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

If $W, X, Y$ and $Z$ are four angles taken in order of a cyclic quadrilateral, then the value of $(13+\cot W+\cot X+\cot Y+\cot Z+13)$ is

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

$\cot \frac{\pi }{16}· \cot \frac{2\pi }{16}· \cot \frac{3\pi }{16}· \cot \frac{4\pi }{16}· \cot \frac{5\pi }{16}· \cot \frac{6\pi }{16}· \cot \frac{7\pi }{16}=$

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The maximum and minimum value of function $f\left(x\right)=\sin 3x+4$ are respectively : 

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

${\left(\frac{\sin 35°}{\cos 55°}\right)}^2+{\left(\frac{\cos 55°}{\sin 35°}\right)}^2-2\cos 30°=$

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The value of $\frac{\tan 330° \sec 420° \sin 300°}{\tan 135° \sin 210° \sec 315°}$ is equal to

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

If $\sec \theta =\frac{13}{12},\theta$ lies in $4^{\text{th }}$quadrant, then $\tan \theta \times \mathrm{cosec}\theta \times \sin \theta \times \cos \theta =$

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

If $\cos \left({\theta }_1\right)+\cos \left({\theta }_2\right)+\cos \left({\theta }_3\right)+\cos \left({\theta }_4\right)=-4,$ then the value of $\cot \left(\frac{{\theta }_1}{2}\right)+\cot \left(\frac{{\theta }_2}{2}\right)+\cot \left(\frac{{\theta }_3}{2}\right)+\cot \left(\frac{{\theta }_4}{2}\right)=$