81.

The value of $\left(1 + \cos \left(\frac{\mathrm{\pi }}{6}\right)\right)\left(1 + \cos \left(\frac{\mathrm{\pi }}{3}\right)\right)\left(1 + \cos \left(\frac{2\mathrm{\pi }}{3}\right)\right)\left(1 + \cos \left(\frac{7\mathrm{\pi }}{6}\right)\right)$ is

• $\frac{3}{16}$

• $\frac{3}{8}$

• $\frac{3}{4}$

• $\frac{1}{2}$

If $\mathrm{P} = \frac{1}{2}{\sin }^2\left(\mathrm{\theta }\right) + \frac{1}{3}{\cos }^2\left(\mathrm{\theta }\right)$ then

• $\frac{1}{3} \le \mathrm{P} \le \frac{1}{2}$

• $\mathrm{P} \ge \frac{1}{2}$

• $2 \le \mathrm{P} \le 3$

• $- \frac{\sqrt{13}}{6} \le \mathrm{P} \le \frac{\sqrt{13}}{6}$

A positive acute angle is divided into two parts whose tangents are $\frac{1}{2}$ and $\frac{1}{3}$. Then, the angle is

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{5}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$

The smallest value of $5\cos \left(\mathrm{\theta }\right) + 12$ is

• 5

• 12

• 7

• 17

Show that

$\frac{\sin \left(\mathrm{\theta }\right)}{\cos \left(3\mathrm{\theta }\right)} + \frac{{\displaystyle \sin \left(3\mathrm{\theta }\right)}}{{\displaystyle \cos \left(9\mathrm{\theta }\right)}} + \frac{{\displaystyle \sin \left(9\mathrm{\theta }\right)}}{{\displaystyle \cos \left(27\mathrm{\theta }\right)}} = \frac{1}{2}\left(\left(\tan 27\mathrm{\theta }\right) - \tan \left(\mathrm{\theta }\right)\right)$

The equation $\sqrt{3}\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right) = 4$ has

• infinitely many solutions

• no solution

• two solutions

• only one solution

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