Trigonometric Ratios & Identities

If $\alpha, \beta$ and $\gamma$ are length of the altitudes of a $\triangle A B C$ with area $\Delta$, then $\frac{\Delta^2}{R^2}\left(\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}\right)$ is equal to

$\sin ^2 \frac{2 \pi}{3}+\cos ^2 \frac{5 \pi}{6}-\tan ^2 \frac{3 \pi}{4}=$

If $f(x)=\sin ^6 x+\cos ^6 x$ for $x \in R$, then $f(x)$ lies in the interval

If $\sin ^4 \theta \cos ^2 \theta=\sum_{n=0}^{\infty} a_{2 n} \cos 2 n \theta$, then the least $n$ for which $a_{2 n}=0$ is

$\sin ^2 5^{\circ}+\sin ^2 10^{\circ}+\sin ^2 15^{\circ}+\ldots+\sin ^2 90^{\circ}$ is equal to

If $A, B, C, D$ are angles of a cyclic quadrilateral, then $\cos A+\cos B+\cos C+\cos D$ is equal to

Match the ranges of the functions given in List - I with those of the items given in List - II

Let $P(\alpha, \beta)$ and $Q(\gamma, \delta)$ be two points that lie on the curve $\tan ^2(x+y)+\cos ^2(x+y)$ $+y^2+2 y=0$ in the $X Y$-plane. If the distance between $P$ and $Q$ is $d$, then $\cos d=$

If $\cos \alpha+\cos \beta=a$ and $\sin \alpha+\sin \beta=b$, then match the items given in List $-A$ with those of their values in List-B

The smallest integer $n$ such that $\frac{1}{\sin 45^{\circ} \sin 46^{\circ}}+\frac{1}{\sin 47^{\circ} \sin 48^{\circ}}+\ldots$ $+\frac{1}{\sin 133^{\circ} \sin 134^{\circ}}=\frac{1}{\sin \left(n^0\right)}$ is

The maximum value of $f(x)=\sin\,(x)$ in the interval$\left[\frac{-\pi\,}{2},\frac{\pi\,}{2}\right]$ is

$\lim _{x \rightarrow 0} \frac{\tan (x)+4 \tan (2 x)-3 \tan (3 x)}{x^2 \tan (x)}$ is equal to