If$\frac{\mathrm{cosec}\theta +\sin \theta }{\mathrm{cosec}\theta -\sin \theta }=\frac{5}{3},$ then what is the value of $\left|\sin \theta \right|?$

Important Questions on Trigonometric Ratios, Functions and Identities

HARD

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

Let $f_k\left(x\right)=\frac{1}{k}\left({\sin }^{k }x+{\cos }^{k }x\right)$ where $x\in R$ and $k\ge 1$. Then $f_4\left(x\right)-f_6\left(x\right)$ equals 

HARD

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

If $x{\sin }^3\theta +y{\cos }^3\theta =\sin \theta \cos \theta$ and $x\sin \theta =y\cos \theta$, then $x^2+y^2$ is

MEDIUM

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

The number of real numbers $\lambda$ for which the equality $\frac{\sin \left(\lambda \alpha \right)}{\sin \alpha }-\frac{\cos \left(\lambda \alpha \right)}{\cos \alpha }=\lambda -1,$ holds for all real $\alpha$ which are not integral multiples of $\frac{\pi }{2}$ is-

EASY

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

Let $P=\left\{\theta :\sin \theta -\cos \theta =\sqrt{2}\cos \theta \right\}$ and $Q=\left\{\theta :\sin \theta +\cos \theta =\sqrt{2}\sin \theta \right\},$ be two sets. Then

HARD

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

$ABCD$ is a trapezium such that $AB$ and $CD$ are parallel and $BC\perp CD$. If $\angle ADB=\theta$, $BC=p$ and $CD=q$, then $AB$ is equal to

HARD

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

If $A>0, B>0$ and $A+B=\frac{\pi }{6}$, then the minimum positive value of $\left(\tan A+\tan B\right)$ is :

EASY

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

If an angle $A$ of a $\mathrm{\Delta }ABC$ satisfies $5\cos A+3=0$, then the roots of the quadratic equation $9x^2+27x+20=0$ are

EASY

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

If $\frac{\sqrt{2}\mathrm{s}\mathrm{i}\mathrm{n}\alpha }{\sqrt{1+\mathrm{c}\mathrm{o}\mathrm{s}2\alpha }}=\frac{1}{7}$ and $\sqrt{\frac{1-\mathrm{c}\mathrm{o}\mathrm{s}2\beta }{2}}=\frac{1}{\sqrt{10}}, \alpha ,\beta \in \left(0,\frac{\pi }{2}\right)$, then $\mathrm{t}\mathrm{a}\mathrm{n}\left(\alpha +2\beta \right)$ is equal to

HARD

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

If $\cos \alpha +\cos \beta =\frac{3}{2} \text{and} \sin \alpha +\sin \beta =\frac{1}{2}$and $\theta$ is the arithmetic mean of $\alpha \& \beta ,$ then $\sin 2\theta +\cos 2\theta$ is equal to:

EASY

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

In a $\mathrm{\Delta }ABC$, $\frac{a}{b}=2+\sqrt{3}$, and $\angle C=60°.$ Then the ordered pair $(\angle A,\angle B)$ is equal to:

HARD

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

The value of ${\sec }^{-1}\left(\frac{1}{4} \sum _{k=0}^{10}\sec \left(\frac{7\pi }{12}+\frac{k\pi }{2}\right)\sec \left(\frac{7\pi }{12}+\frac{\left(k+1\right)\pi }{2}\right)\right)$ in the interval $\left[-\frac{\pi }{4},\frac{3\pi }{4}\right]$ equals

HARD

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

If $\mathrm{s}\mathrm{i}{\mathrm{n}}^4\alpha +4\mathrm{c}\mathrm{o}{\mathrm{s}}^4\beta +2=4\sqrt{2}\mathrm{s}\mathrm{i}\mathrm{n}\alpha \mathrm{c}\mathrm{o}\mathrm{s}\beta$, $\alpha , \beta \in \left[0,\mathrm{\pi }\right]$, then $\mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha +\beta \right)-\mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha -\beta \right)$ is equal to

EASY

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

The value of $\mathrm{s}\mathrm{i}\mathrm{n}10°\mathrm{s}\mathrm{i}\mathrm{n}30°\mathrm{s}\mathrm{i}\mathrm{n}50°\mathrm{s}\mathrm{i}\mathrm{n}70°$ is:

MEDIUM

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

If $\mathrm{cosec} \mathrm{\theta }=\frac{\mathrm{p}+\mathrm{q}}{ \mathrm{p}-\mathrm{q} }\left(\mathrm{p}\ne \mathrm{q}, \mathrm{p}\ne 0\right),$then $\left|\cot \left(\frac{\mathrm{\pi }}{4}+\frac{\mathrm{\theta }}{2}\right)\right|$ is equals to:

EASY

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

If $\mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha +\beta \right)=\frac{3}{5}\mathrm{ },\sin (\alpha -\beta )=\frac{5}{13}$ and $0<\alpha ,\beta <\frac{\pi }{4},$ then $\tan \left(2\alpha \right)$ is equal to:

EASY

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

The equation $y=\sin x·\sin \left(x+2\right)-{\sin }^2(x+1)$ represents a straight line lying in:

MEDIUM

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

The value of ${\cos }^{2}10°–\cos 10°\mathrm{ }\cos 50°+\mathrm{c}\mathrm{o}{\mathrm{s}}^{2}50°$ is

HARD

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

The integer part of the number $\sum _{\mathrm{k}=0}^{44}\frac{1}{\mathrm{cosk}°\cos \left(\mathrm{k}+1\right)°}$ is $\left(\sin 1°=0.0174524\right)$

MEDIUM

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

The expression $\frac{\mathrm{tanA}}{1-\mathrm{cotA}}+\frac{\mathrm{cotA}}{1-\mathrm{tanA}}$ can be written as :

HARD

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

The value of $\sum _{k=1}^{13}\frac{1}{\sin \left(\frac{\pi }{4}+\frac{\left(k-1\right)\pi }{6}\right)\sin \left(\frac{\pi }{4}+\frac{k\pi }{6}\right)}$ is equal to