$$\begin{gathered} \mathrm{If} \left(1 + \tan \left(\mathrm{\alpha }\right)\right)\left(1 + \tan \left(4\mathrm{\alpha }\right)\right) = 2, \mathrm{\alpha } \in \left(0, \frac{\mathrm{\pi }}{16}\right), \\ \mathrm{then} \mathrm{\alpha } = ? \end{gathered}$$

• $\frac{\mathrm{\pi }}{20}$

• $\frac{\mathrm{\pi }}{30}$

• $\frac{\mathrm{\pi }}{40}$

• $\frac{\mathrm{\pi }}{50}$

$\mathrm{If} \cos \left(\mathrm{\theta }\right) = \frac{\cos \left(\mathrm{\alpha }\right) - \cos \left(\mathrm{\beta }\right)}{1 - \cos \left(\mathrm{\alpha }\right)\cos \left(\mathrm{\beta }\right)}, \mathrm{then} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{values} \mathrm{of} \tan \left(\frac{\mathrm{\theta }}{2}\right) \mathrm{is}$

• $\cot \left(\frac{\mathrm{\beta }}{2}\right)\tan \left(\frac{\mathrm{\alpha }}{2}\right)$

• $\tan \left(\frac{\mathrm{\alpha }}{2}\right)\tan \left(\frac{\mathrm{\beta }}{2}\right)$

• $\tan \left(\frac{\mathrm{\beta }}{2}\right)\cot \left(\frac{\mathrm{\alpha }}{2}\right)$

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

The value of the expression $\frac{1 + \sin \left(2\mathrm{\alpha }\right)}{\cos \left(2\mathrm{\alpha } - 2\mathrm{\pi }\right)\tan \left(\mathrm{\alpha } - {\displaystyle \frac{3\mathrm{\pi }}{4}}\right)} - \frac{1}{4}\sin \left(2\mathrm{\alpha }\right)\left[\cot \left(\frac{\mathrm{\alpha }}{2}\right) + \cot \left(\frac{3\mathrm{\pi }}{2} + \frac{\mathrm{\alpha }}{2}\right)\right] \mathrm{is}$

• 0

• 1

• ${\sin }^2\left(\frac{\mathrm{\alpha }}{2}\right)$

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

$\mathrm{If} \frac{1}{6}\sin \left(\mathrm{\theta }\right), \cos \left(\mathrm{\theta }\right) \mathrm{and} \tan \left(\mathrm{\theta }\right) \mathrm{are} \mathrm{in} \mathrm{geometric} \mathrm{progression}, \mathrm{then} \mathrm{the} \mathrm{solution} \mathrm{set} \mathrm{of} \mathrm{\theta } \mathrm{is}$

• $2\mathrm{n\pi } \pm \left(\frac{\mathrm{\pi }}{6}\right)$

• $2\mathrm{n\pi } \pm \left(\frac{\mathrm{\pi }}{3}\right)$

• $\mathrm{n\pi } + {\left( - 1\right)}^{\mathrm{n}}\left(\frac{\mathrm{\pi }}{3}\right)$

• $\mathrm{n\pi } + \left(\frac{\mathrm{\pi }}{3}\right)$

$$\begin{gathered} \mathrm{In} ∆\mathrm{ABC} \mathrm{if} \mathrm{x} = \tan \left(\frac{\mathrm{B} - \mathrm{C}}{2}\right)\tan \left(\frac{\mathrm{A}}{2}\right), \mathrm{y} = \tan \left(\frac{\mathrm{C} - \mathrm{A}}{2}\right)\tan \left(\frac{\mathrm{B}}{2}\right) \mathrm{and} \\ \mathrm{z} = \tan \left(\frac{\mathrm{A} - \mathrm{B}}{2}\right)\tan \left(\frac{\mathrm{C}}{2}\right), \mathrm{then} \left(\mathrm{x} + \mathrm{y}\hspace{0.17em}+ \mathrm{z}\right) = ? \end{gathered}$$

• xyz

• - xyz

• 2xyz

• $\frac{1}{2}\mathrm{xyz}$

$\mathrm{If} \mathrm{A} > 0, \mathrm{B} > 0 \mathrm{and} \mathrm{A} + \mathrm{B} = \frac{\mathrm{\pi }}{3}, \mathrm{then} \mathrm{the} \mathrm{maximum} \mathrm{value} \mathrm{of} \mathrm{Atan}\left(\mathrm{B}\right) \mathrm{is}$

• $\frac{1}{\sqrt{3}}$

• $\frac{1}{3}$

• $\frac{1}{2}$

• $\sqrt{3}$

$\mathrm{In} ∆\mathrm{ABC}, \mathrm{if} \mathrm{bcos}\left(\mathrm{\theta }\right) = \mathrm{c} - \mathrm{a}, (\mathrm{where} \mathrm{\theta } \mathrm{is} \mathrm{an} \mathrm{acute} \mathrm{angle}), \mathrm{then} (\mathrm{c} - \mathrm{a}) \tan \left(\mathrm{\theta }\right) = ?$

• $2\sqrt{\mathrm{ca}}\cos \left(\frac{\mathrm{B}}{2}\right)$

• $2\sqrt{\mathrm{ac}}\sin \left(\frac{\mathrm{B}}{2}\right)$

• $2\mathrm{cacos}\left(\frac{\mathrm{B}}{2}\right)$

• $\text{2casin}\left(\frac{\mathrm{B}}{2}\right)$

$$\begin{gathered} \mathrm{If} \tan \left(\mathrm{\alpha }\right) \mathrm{and} \tan \left(\mathrm{\beta }\right) \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation} {\mathrm{x}}^2 + \mathrm{px} + \mathrm{q} = 0, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \\ {\sin }^2\left(\mathrm{\alpha } + \mathrm{\beta }\right) + \mathrm{pcos}\left(\mathrm{\alpha } + \mathrm{\beta }\right)\sin \left(\mathrm{\alpha } + \mathrm{\beta }\right) + {\mathrm{qcos}}^2\left(\mathrm{\alpha } + \mathrm{\beta }\right) \end{gathered}$$

• p + q

• p

• q

• $\frac{\mathrm{p}}{\mathrm{p} + \mathrm{q}}$

$1 + \cos \left(10°\right) \hspace{0.17em}+ \cos \left(20°\right) + \cos \left(30°\right) = ?$

• $4\sin \left(10°\right)\sin \left(20°\right)\sin \left(30°\right)$

• $4\cos \left(5°\right)\cos \left(10°\right)\cos \left(15°\right)$

• $4\cos \left(10°\right)\cos \left(20°\right)\cos \left(30°\right)$

• $4\sin \left(5°\right)\sin \left(10°\right)\sin \left(15°\right)$

$\mathrm{The} \mathrm{value} \mathrm{of} {\left(\frac{1 + \sin \left({\displaystyle \frac{2\mathrm{\pi }}{9}}\right) + \mathrm{icos}\left(\frac{2\mathrm{\pi }}{9}\right)}{1 + \sin \left(\frac{2\mathrm{\pi }}{9}\right) - \mathrm{icos}\left(\frac{2\mathrm{\pi }}{9}\right)}\right)}^3 \mathrm{is} :$

• $- \frac{1}{2}\left(\sqrt{3} - \mathrm{i}\right)$

• $\frac{1}{2}\left(\sqrt{3} - \mathrm{i}\right)$

• $\frac{1}{2}\left(1 - \mathrm{i}\sqrt{3}\right)$

• $- \frac{1}{2}\left(1 - \mathrm{i}\sqrt{3}\right)$