The value of

$\tan \left(\mathrm{\alpha }\right) + 2\tan \left(2\mathrm{\alpha }\right) + 4\tan \left(4\mathrm{\alpha }\right) + ... + 2^{\mathrm{n} - 1}\tan \left(2^{\mathrm{n} - 1}\mathrm{\alpha }\right) + 2^{\mathrm{n}}\cot \left(2^{\mathrm{n}}\mathrm{\alpha }\right)$ is

• $\cot \left(2^{\mathrm{n}}\mathrm{\alpha }\right)$

• $2^{\mathrm{n}}\tan \left(2^{\mathrm{n}}\mathrm{\alpha }\right)$

• 0

• $\cot \left(\mathrm{\alpha }\right)$

If $\tan \left(\frac{\mathrm{\alpha \pi }}{4}\right) = \cot \left(\frac{\mathrm{\beta \pi }}{4}\right)$, then

• $\mathrm{\alpha } + \mathrm{\beta } = 0$

• $\mathrm{\alpha } + \mathrm{\beta } = 2\mathrm{n}$

• $\mathrm{\alpha } + \mathrm{\beta } = 2\mathrm{n} + 1$

• $\mathrm{\alpha } + \mathrm{\beta } = 2(2\mathrm{n} + 1), \forall \mathrm{n} \mathrm{is} \mathrm{an} \mathrm{integer}$

The principal value of ${\sin }^{-1}\left(\tan \left(- \frac{5\mathrm{\pi }}{4}\right)\right)$ is

• $\frac{\mathrm{\pi }}{4}$

• - $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $- \frac{\mathrm{\pi }}{2}$

The value of $\cos \left(\frac{\mathrm{\pi }}{15}\right)\cos \left(\frac{2\mathrm{\pi }}{15}\right)\cos \left(\frac{4\mathrm{\pi }}{15}\right)\cos \left(\frac{8\mathrm{\pi }}{15}\right)$

• $\frac{1}{16}$

• $- \frac{1}{16}$

• 1

• 0

The principal amplitude of ${\left(\sin \left(40°\right) + \mathrm{icos}\left(40°\right)\right)}^5$

• 70$°$

• - 110$°$

• 110°

• - 70° 

Find the general solution of $\sec \left(\mathrm{\theta }\right) + 1 = \left(2 + \sqrt{3}\right)\tan \left(\mathrm{\theta }\right)$

The real part of ${\left(1 - \cos \left(\mathrm{\theta }\right) + 2\mathrm{isin}\left(\mathrm{\theta }\right)\right)}^{- 1}$

• $\frac{1}{3 + 5\cos \left(\mathrm{\theta }\right)}$

• $\frac{1}{5 - 3\cos \left(\mathrm{\theta }\right)}$

• $\frac{1}{3 - 5\cos \left(\mathrm{\theta }\right)}$

• $\frac{1}{5 + 3\cos \left(\mathrm{\theta }\right)}$

The value of $\sin \left(36°\right)\sin \left(72°\right)\sin \left(108°\right)\sin \left(144°\right)$ is equal to

• $\frac{1}{4}$

• $\frac{1}{16}$

• $\frac{3}{4}$

• $\frac{5}{16}$

If $\sin \left(\mathrm{A}\right) = \frac{1}{\sqrt{10}} \mathrm{and} \sin \left(\mathrm{B}\right) = \frac{1}{\sqrt{5}}$ where A and B are positive acute angles, then A + B is equal to 

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

The expression ${\tan }^2\left(\mathrm{\alpha }\right) + {\cot }^2\left(\mathrm{\alpha }\right)$ is

• $\ge 2$

• $\le 2$

• $\ge - 2$

• None of these