$$ \tan ^2 \frac{\pi}{16}+\tan ^2 \frac{2 \pi}{16}+\tan ^2 \frac{3 \pi}{16}+\tan ^2 \frac{4 \pi}{16} $$

$+\tan ^2 \frac{5 \pi}{16}+\tan ^2 \frac{6 \pi}{16}+\tan ^2 \frac{7 \pi}{16}$ is equal to

$$ \begin{aligned} & \sin ^2 18^{\circ}+\sin ^2 24^{\circ}+\sin ^2 36^{\circ}+\sin ^2 42^{\circ}+\sin ^2 78^{\circ} \\ & +\sin ^2 90^{\circ}+\sin ^2 96^{\circ}+\sin ^2 102^{\circ}+\sin ^2 138^{\circ}+\sin ^2 162^{\circ} \text { is } \\ & \text { equal to } \end{aligned} $$

If $\sinh x=\frac{\sqrt{21}}{2}$, then $\cosh 2 x+\sinh 2 x$ is equal to

If $M_1$ and $M_2$ are the maximum values of $\frac{1}{11 \cos 2 x+60 \sin 2 x+69}$ and $3 \cos ^2 5 x+4 \sin ^2 5 x$ respectively, then $\frac{M_1}{M_2}=$

$$ 4 \cos \frac{\pi}{7} \cos \frac{\pi}{5} \cos \frac{2 \pi}{7} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{7}= $$

Assertion (A) : If $A=10^{\circ}, B=16^{\circ}$ and $C=19^{\circ}$, then $\tan 2 A \tan 2 B+\tan 2 B \tan 2 C+\tan 2 C \tan 2 A=1$

Reason (R) : If $A+B+C=180^{\circ}, \cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}$

$$ =\cot \frac{A}{2} \cot \frac{B}{2} \cot \frac{C}{2} $$

Which of the following is correct ?

Statement $(\mathrm{S} 1) \sin 55^{\circ}+\sin 53^{\circ}-\sin 19^{\circ}-\sin 17^{\circ}=\cos 2^{\circ}$

Statement (S2) Range of $\frac{1}{3-\cos 2 x}$ is $\left[\frac{1}{4}, \frac{1}{2}\right]$

Which one of the following is correct?

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

If $$\sin \theta=-\frac{3}{4}$$, then $$\sin 2 \theta=$$

$$\begin{aligned} & \frac{1}{\sin 1^{\circ} \sin 2^{\circ}}+\frac{1}{\sin 2^{\circ} \sin 3^{\circ}}+\ldots +\frac{1}{\sin 89^{\circ}+\sin 90^{\circ}}= \end{aligned}$$

Which of the following trigonometric values are negative?

I. $$\sin \left(-292^{\circ}\right)$$

II. $$\tan \left(-190^{\circ}\right)$$

III. $$\cos \left(-207^{\circ}\right)$$

IV. $$\cot \left(-222^{\circ}\right)$$

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

A true statement among the following identities is

If $$A+B+C=\pi, \cos B=\cos A \cos C$$, then $$\tan A \tan C=$$

The value of $$\tan \left(\frac{7 \pi}{8}\right)$$ is

$$1+\sec ^2 x \sin ^2 x=$$

If the identity $$\cos ^4 \theta=a \cos 4 \theta+b \cos 2 \theta+c$$ holds for some $$a, b, c \in Q$$ then $$(a, b, c)=$$

The value of $$\frac{\sin \theta+\sin 3 \theta}{\cos \theta+\cos 3 \theta}$$ is

If $$(1+\tan 1^{\circ})(1+\tan 2^{\circ}) \ldots(1+\tan 45^{\circ})=2^n,$$ then $$n=$$

$$\frac{\cos \theta}{1-\tan \theta}+\frac{\sin \theta}{1-\cot \theta}=$$

If $$\operatorname{cosech} x=\frac{4}{5}$$, then $$\sinh x=$$

What is the value of $$\cos \left(22 \frac{1}{2}\right)^{\circ}$$ ?

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

If $$\tan \beta=\frac{\tan \alpha+\tan \gamma}{1+\tan \alpha \tan \gamma}$$, then $$\frac{\sin 2 \alpha+\sin 2 \gamma}{1+\sin 2 \alpha \sin 2 \gamma}$$ is equal to

The sides of a triangle inscribed in a given circle subtend angles $$\alpha, \beta, \gamma$$ at the center. The minimum value of the AM of $$\cos \left(\alpha+\frac{\pi}{2}\right), \cos \left(\beta+\frac{\pi}{2}\right)$$ and $$\cos \left(\gamma+\frac{\pi}{2}\right)$$ is equal to

In a $$\triangle A B C$$, if $$3 \sin A+4 \cos B=6$$ and $$4 \sin B+3 \cos A=1$$, then $$\sin (A+B)$$ is equal to

$$\tan \alpha+2 \tan 2 \alpha+4 \tan 4 \alpha+8 \cot 8 \alpha$$ is equal to

If $$f(x)=\frac{\cot x}{1+\cot x}$$ and $$\alpha+\beta=\frac{5 \pi}{4}$$, then the value of $$f(\alpha) f(\beta)$$ is equal to

In $$\triangle A B C \cdot \frac{a+b+c}{B C+A B}+\frac{a+b+c}{A C+A B}=3$$, then $$\tan \frac{C}{8}$$ is equal to

Mean of the values $$\sin ^2 10 Y, \sin ^2 20 Y, \sin ^2 30 Y, \ldots \ldots \ldots ., \sin ^2 90 Y$$ is

When the coordinate axes are rotated through an angle 135$$\Upsilon$$, the coordinates of a point $$P$$ in the new system are known to be $$(4,-3)$$. Then find the coordinates of $$P$$ in the original system.

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

$$\tan 2 \alpha \cdot \tan (30 Y-\alpha)+\tan 2 \alpha \cdot \tan (60 Y-\alpha)+\tan (60 \Upsilon-\alpha) \cdot \tan (30 \gamma-\alpha)$$ is equal to

If $$\sin \alpha - \cos \alpha = m$$ and $$\sin 2\alpha = n - {m^2}$$, where $$ - \sqrt 2 \le m \le \sqrt 2 $$, then n is equal to

If $$\sinh u=\tan \theta$$, then $$\cosh u$$ is equal to