Sum of Sines of Angles in Arithmetic Progression

Q. The sum of the infinite series
${\sin }^{-1}\frac{1}{\sqrt{2}}+{\sin }^{-1}\left(\frac{\sqrt{2}-1}{\sqrt{6}}\right)+{\sin }^{-1}\left(\frac{\sqrt{3}-\sqrt{2}}{\sqrt{12}}\right)+\cdots +{\sin }^{-1}\left(\frac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n(n+1)}}\right)+\cdots$ is

1. $\frac{\pi }{4}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{2}$
4. $\pi$

Q. The average value of $\sin 2°,\sin 4°,\sin 6°,...,\sin 180°\mathrm{is}$

1. 
   $\frac{1}{90}\cot 1°$
2. 
   $\frac{1}{90}\tan 1°$
3. 
   $\frac{1}{90}\tan 89°$
4. 
   $\frac{1}{90}\cot 89°$

Q. Let common ratio of a G.P. be $\cot \theta$ and sum of its infinite number of terms is $10$. A new G.P. is formed by taking cube of each of the terms of given series. If sum of infinite terms of new series is $\frac{1000}{7}$, then the number of possible value(s) of $\theta \in [0,5\pi ]$ is

Q. If $\sin \frac{\pi }{14}\sin \frac{3\pi }{14}\sin \frac{5\pi }{14}\sin \frac{7\pi }{14}\sin \frac{9\pi }{14}\sin \frac{11\pi }{14}\sin \frac{13\pi }{14}=\frac{1}{k},$ then $k$ is equal to

Q.

If $S=\sin \frac{\pi }{n}+\sin \frac{3\pi }{n}+\sin \frac{5\pi }{n}+\cdot \cdot \cdot n$ terms.
Then the value $nS$ is

Q. The sum of the series ${\cot }^{-1}(2^2+\frac{1}{2})+{\cot }^{-1}(2^3+\frac{1}{2^2})+{\cot }^{-1}(2^4+\frac{1}{2^3})+......\infty$ is equal to

1. ${\tan }^{-1}\frac{1}{2}$
2. ${\tan }^{-1}2$
3. ${\tan }^{-1}2+{\tan }^{-1}3$
4. ${\tan }^{-1}\frac{1}{2}+{\cot }^{-1}\frac{1}{2}$

Q. If $\sin \frac{\pi }{14}\sin \frac{3\pi }{14}\sin \frac{5\pi }{14}\sin \frac{7\pi }{14}\sin \frac{9\pi }{14}\sin \frac{11\pi }{14}\sin \frac{13\pi }{14}=\frac{1}{k},$ then $k$ is equal to

Q.

If $S=\sin \frac{\pi }{n}+\sin \frac{3\pi }{n}+\sin \frac{5\pi }{n}+\cdot \cdot \cdot n$ terms.
Then the value $nS$ is

Q. The average of $m\sin \left(m^{\circ }\right)$ where $m=2,4,6,\cdots ,180$ is equal to
(correct answer + 1, wrong answer - 0.25)

1. $\cot {89}^{\circ }$
2. $\cot 1^{\circ }$
3. $90\sin {89}^{\circ }$
4. $\tan 1^{\circ }$

Q. The average value of $\sin 2°,\sin 4°,\sin 6°,...,\sin 180°\mathrm{is}$

1. 
   $\frac{1}{90}\cot 1°$
2. 
   $\frac{1}{90}\tan 1°$
3. 
   $\frac{1}{90}\tan 89°$
4. 
   $\frac{1}{90}\cot 89°$

Q. State true or false:

1. True
2. False