Mathematics : Trigonometric Functions

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

The equation $3{\sin }^2\left(\mathrm{x}\right) + 10\cos \left(\mathrm{x}\right) - 6$ is satisfied, if

• $\mathrm{x} = \mathrm{n\pi } \pm {\cos }^{-1}\left(\frac{1}{3}\right)$

• $\mathrm{x} = 2\mathrm{n\pi } \pm {\cos }^{-1}\left(\frac{1}{3}\right)$

• $\mathrm{x} = \mathrm{n\pi } \pm {\cos }^{-1}\left(\frac{1}{6}\right)$

• $\mathrm{x} = 2\mathrm{n\pi } \pm {\cos }^{-1}\left(\frac{1}{6}\right)$

If ${\mathrm{e}}^{\cos \left(\mathrm{x}\right)} - {\mathrm{e}}^{- \cos \left(\mathrm{x}\right)} = 4$, then the value of $\cos \left(\mathrm{x}\right)$ is

• $\log \left(2 + \sqrt{5}\right)$

• $- \log \left(2 + \sqrt{5}\right)$

• $\log \left(- 2 + \sqrt{5}\right)$

• None of these

If ${\tan }^{-1}\left(\frac{\mathrm{a}}{\mathrm{x}}\right) + {\tan }^{-1}\left(\frac{\mathrm{b}}{\mathrm{x}}\right) = \frac{\mathrm{\pi }}{2}$, then x is equal to

• $\sqrt{\mathrm{ab}}$

• $\sqrt{2\mathrm{ab}}$

• 2ab

• ab

If $\cos \left(\mathrm{\theta } - \mathrm{\alpha }\right) = \mathrm{a}, \cos \left(\mathrm{\theta } - \mathrm{\beta }\right) = \mathrm{b}$, then

${\sin }^2\left(\mathrm{\alpha } - \mathrm{\beta }\right) + 2\mathrm{abcos}\left(\mathrm{\alpha } - \mathrm{\beta }\right)$ is equal to

• a² + b²

• a² - b²

• b² - a²

• - a² - b²

The most general solutions of the equation

$\sec \left(\mathrm{x}\right) - 1 = \left(\sqrt{2} - 1\right)\tan \left(\mathrm{x}\right)$ are given by

• $\mathrm{n\pi } + \frac{\mathrm{\pi }}{8}$

• $2\mathrm{n\pi }, 2\mathrm{n\pi } + \frac{\mathrm{\pi }}{4}$

• $2\mathrm{n\pi }$

• None of these

If $\mathrm{\alpha } + \mathrm{\beta } - \mathrm{\gamma } = \mathrm{\pi }, \mathrm{then} {\sin }^2\left(\mathrm{\alpha }\right) + {\sin }^2\left(\mathrm{\beta }\right) - {\sin }^2\left(\mathrm{\gamma }\right)$ is equal to

• $2\sin \left(\mathrm{\alpha }\right)\sin \left(\mathrm{\beta }\right)\cos \left(\mathrm{\gamma }\right)$

• $2\cos \left(\mathrm{\alpha }\right)\cos \left(\mathrm{\beta }\right)\cos \left(\mathrm{\gamma }\right)$

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

• None of the above

The range of the function f (x) = ${}^{7 - \mathrm{x}}\mathrm{P}_{\mathrm{x} - 3}$ is

• {1, 2, 3}

• {1, 2, 3, 4, 5, 6}

• {1, 2, 3, 4}

• {1, 2, 3, 4, 5}