$\mathrm{What} \mathrm{is} \frac{1}{\sin \left(10°\right)} - \frac{\sqrt{3}}{\cos \left(10°\right)} = ?$

• 0

• 1

• 2

• 4

$\mathrm{If} \mathrm{k} = \sin \left(\frac{\mathrm{\pi }}{18}\right)\sin \left(\frac{5\mathrm{\pi }}{18}\right)\sin \left(\frac{7\mathrm{\pi }}{18}\right), \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{k} ?$

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{8}$

• $\frac{1}{16}$

$\mathrm{The} \mathrm{expression} \frac{\sin \left(\mathrm{\alpha }\right) + \sin \left(\mathrm{\beta }\right)}{\cos \left(\mathrm{\alpha }\right) + \cos \left(\mathrm{\beta }\right)} = ?$

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

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

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

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

What is the value of $\tan \left(18°\right)$ ?

• $\frac{\sqrt{5} - 1}{\sqrt{10 +\hspace{0.17em}2\sqrt{5}}}$

• $\frac{\sqrt{5} - 1}{\sqrt{10 + \sqrt{5}}}$

• $\frac{\sqrt{10 + 2\sqrt{5}}}{\sqrt{5} - 1}$

• $\frac{\sqrt{10 + \sqrt{5}}}{\sqrt{5} - 1}$

$\mathrm{If} \tan \left(\mathrm{\alpha } + \mathrm{\beta }\right) = 2 \mathrm{and} \tan \left(\mathrm{\alpha } + \mathrm{\beta }\right) = 1, \mathrm{then} \tan \left(2\mathrm{\alpha }\right) = ?$

•  - 3

•  - 2

• $- \frac{1}{3}$

• 1

Consider the following for triangle ABC :

$$\begin{gathered} \mathrm{I}. \sin \left(\frac{\mathrm{B} + \mathrm{C}}{2}\right) = \cos \left(\frac{\mathrm{A}}{2}\right) \\ \mathrm{II}. \tan \left(\frac{\mathrm{B} + \mathrm{C}}{2}\right) = \cot \left(\frac{\mathrm{A}}{2}\right) \\ \mathrm{III}. \sin \left(\mathrm{B} +\hspace{0.17em}\mathrm{C}\right) = \sin \left(\mathrm{A}\right) \\ \mathrm{IV}. \tan \left(\mathrm{B} +\mathrm{C}\right) = - \tan \left(\mathrm{A}\right) \end{gathered}$$

Which of the above are correct ?

• 1 and 3

• 1 and 2

• 1 and 4

• 2 and 3

$\mathrm{If} \sin \left(\mathrm{\theta }\right) = 3\sin (\mathrm{\theta } + 2\mathrm{\alpha }), \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \tan (\mathrm{\theta } + \mathrm{\alpha })+ 2\tan \left(\mathrm{\alpha }\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

•  - 1

• 0

• 1

• 2

$\mathrm{If} \sec \left(\mathrm{\theta }\right) - \csc \left(\mathrm{\theta }\right) = \frac{4}{3}, \mathrm{then} \mathrm{what} \mathrm{is} \left(\sin \left(\mathrm{\theta }\right) - \cos \left(\mathrm{\theta }\right)\right) = ?$

•  - 2 only

• $\frac{1}{2} \mathrm{only}$

• $\mathrm{Both} - 2 \mathrm{and} \frac{1}{2}$

• Neither $\frac{1}{2}$ nor  - 2

49.

The value of tan(9)° - tan(27)° - tan(63)° + tan(81)° is equal to :

•  - 1

• 0

• 1

• 4

$\mathrm{The} \mathrm{value} \mathrm{of} \sqrt{3}\csc \left(20°\right) - \sec \left(20°\right) = ?$

• 4

• 2

• 1

• - 4