Trigonometric Ratios & Identities

$\cos ^{2}\left(\frac{\pi}{6}+\theta\right)-\sin ^{2}\left(\frac{\pi}{6}-\theta\right)=$

A $10$inches long pencil $AB$ with mid point $C$ and a small eraser $P$ are placed on the horizontal top of a table such that $PC=\sqrt{5}$ inches and $\angle\,PCB=\tan\,^{-1}(2)$. The acute angle through which the…

$\alpha\,=\sin\,36^{\circ}$is a root of which of the following equation

Let$S=\left{\theta\,\in\,\left(0,\frac{\pi\,}{2}\right):\underset{m=1}{\overset{9}{\sum\,}}\sec\,\left(\theta\,+\left(m-1\right)\frac{\pi\,}{6}\right)\sec\,\left(\theta\,+\frac{m\pi\,}{6}\right)=-\frac{8}{\sqrt{3}}\right…

If $\cot\,\alpha\,=1$and$\sec\,\beta\,=-\frac{5}{3}$, where $\pi\,<\alpha\,<\frac{3\pi\,}{2}$and $\frac{\pi\,}{2}<\beta\,<\pi\,$, then the value of $\tan\,\left(\alpha\,+\beta\,\right)$and the quadrant in which…

$\operatorname{cosec}\,18^{\circ}$ is a root of the equation:

The maximum value of $3\cos\,\theta\,+5\sin\,\left(\theta\,-\frac{\pi\,}{6}\right)$ for any real value of $\theta\,$ is :

For $\alpha\,,\beta\,\in\,\left(0,\frac{\pi\,}{2}\right)$ let $3\sin\,(\alpha\,+\beta\,)=2\sin\,(\alpha\,-\beta\,)$ and a real number $k$ be such that $\tan\,\alpha\,=\tan\,\beta\,$. Then the value of$k$ is equal to

The number of integral values of $k$for which the equation $3\sin\,x+4\cos\,x=k+1$ has a solution,$k\in\,R$ is _______?

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

If $15\sin\,^{4}\alpha\,+10\cos\,^{4}\alpha\,=6,$ for some $\alpha\,\in\,R,$thenthe value of $27\sec\,^{6}\alpha\,+8\operatorname{cosec}\,^{6}\alpha\,$ is equal to :

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