What is the value of $\frac{\sin \left(34°\right)\cos \left(236°\right) - \sin \left(56°\right)\sin \left(124°\right)}{\cos \left(28°\right)\cos \left(88°\right) + \cos \left(178°\right)\sin \left(208°\right)}$ = ?

•  - 2

•  - 1

• 2

• 1

$\tan \left(54°\right) \mathrm{can} \mathrm{be} \mathrm{expressed} \mathrm{as}$

• $\frac{\sin \left(9°\right) + \cos \left(9°\right)}{\sin \left(9°\right) - \cos \left(9°\right)}$

• $\frac{\sin \left(9°\right) - \cos \left(9°\right)}{\sin \left(9°\right) + \cos \left(9°\right)}$

• $\frac{\cos \left(9°\right) + \sin \left(9°\right)}{\cos \left(9°\right) - \sin \left(9°\right)}$

• $\frac{\sin \left(36°\right)}{\cos \left(36°\right)}$

If p = $\mathrm{Xcos}\left(\mathrm{\theta }\right) - \mathrm{Ysin}\left(\mathrm{\theta }\right)$, q = $\mathrm{Xsin}\left(\mathrm{\theta }\right) + \mathrm{Ycos}\left(\mathrm{\theta }\right)$ and p² + 4pq + q² = AX² + BY², $0 \le \mathrm{\theta } \le \frac{\mathrm{\pi }}{2}$. What is the value of $\mathrm{\theta }$ ?

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

$$\begin{gathered} \mathrm{If} \mathrm{p} = \left(\mathrm{Xcos}\left(\mathrm{\theta }\right) - \mathrm{Ysin}\left(\mathrm{\theta }\right)\right), \mathrm{q} = \mathrm{Xsin}\left(\mathrm{\theta }\right) + \mathrm{Ycos}\left(\mathrm{\theta }\right) \mathrm{and} \\ {\mathrm{p}}^2 + 4\mathrm{pq} + {\mathrm{q}}^2 = {\mathrm{AX}}^2 + {\mathrm{BY}}^2, 0 \le \mathrm{\theta } \le \frac{\mathrm{\pi }}{2} \end{gathered}$$

What is the value of A ?

• 4

• 3

• 2

• 1

$$\begin{gathered} \mathrm{If} \mathrm{p} = \left(\mathrm{Xcos}\left(\mathrm{\theta }\right) - \mathrm{Ysin}\left(\mathrm{\theta }\right)\right), \mathrm{q} = \mathrm{Xsin}\left(\mathrm{\theta }\right) + \mathrm{Ycos}\left(\mathrm{\theta }\right) \mathrm{and} \\ {\mathrm{p}}^2 + 4\mathrm{pq} + {\mathrm{q}}^2 = {\mathrm{AX}}^2 + {\mathrm{BY}}^2, 0 \le \mathrm{\theta } \le \frac{\mathrm{\pi }}{2} \end{gathered}$$

• -1

• 0

• 1

• 2

$$\begin{gathered} \mathrm{It} \mathrm{is} \mathrm{given} \mathrm{that} \cos \left(\mathrm{\theta } - \mathrm{\alpha }\right) = \mathrm{a}, \cos \left(\mathrm{\theta } - \mathrm{\beta }\right) = \mathrm{b} \\ \mathrm{What} \mathrm{is} \cos \left(\mathrm{\alpha } - \mathrm{\beta }\right) = ? \end{gathered}$$ 

• $\mathrm{ab} + \sqrt{1 - {\mathrm{a}}^2}\sqrt{1 - {\mathrm{b}}^2}$

• $\mathrm{ab} - \sqrt{1 - {\mathrm{a}}^2}\sqrt{1 - {\mathrm{b}}^2}$

• $\mathrm{a}\sqrt{1 - {\mathrm{b}}^2} - \mathrm{b}\sqrt{1 - {\mathrm{a}}^2}$

• $\mathrm{a}\sqrt{1 - {\mathrm{b}}^2} + \mathrm{b}\sqrt{1 - {\mathrm{a}}^2}$

It is given that $\cos \left(\mathrm{\theta } - \mathrm{\alpha }\right) = \mathrm{a}, \cos \left(\mathrm{\theta } - \mathrm{\beta }\right) = \mathrm{b}.$ $\mathrm{What} \mathrm{is} {\sin }^2\left(\mathrm{\alpha } - \mathrm{\beta }\right) + 2\mathrm{abcos}\left(\mathrm{\alpha } - \mathrm{\beta }\right) = ?$

• a² + b²

• a² - b²

• b² - a²

• - (a² + b²)

$\mathrm{If} \sin \left(\mathrm{\alpha }\right) + \cos \left(\mathrm{\alpha }\right) = \mathrm{p}, \mathrm{then} \mathrm{what} \mathrm{is} {\cos }^2\left(2\mathrm{\alpha }\right) = ?$

• ${\mathrm{p}}^2$

• ${\mathrm{p}}^2 - 1$

• ${\mathrm{p}}^2\left(2 - {\mathrm{p}}^2\right)$

• ${\mathrm{p}}^2 + 1$

$\mathrm{If} \tan \left(\mathrm{\theta }\right) = \frac{1}{2} \mathrm{and} \tan \left(\mathrm{\phi }\right) = \frac{1}{3}, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} \left(\mathrm{\theta } + \mathrm{\varphi }\right)$

• 0

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

10.

$\mathrm{If} \cos \left(\mathrm{A}\right) = \frac{3}{4}, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} \sin \left(\frac{\mathrm{A}}{2}\right)\sin \left(\frac{3\mathrm{A}}{2}\right) = ?$

• $\frac{5}{8}$

• $\frac{5}{16}$

• $\frac{5}{24}$

• $\frac{7}{32}$