Find the value of $\sin \left(12°\right)\sin \left(48°\right)\sin \left(54°\right)$.

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{6}$

• $\frac{1}{8}$

If $3\sin \left(\mathrm{\theta }\right) + 5\cos \left(\mathrm{\theta }\right)$, then the value of $5\sin \left(\mathrm{\theta }\right) - 3\cos \left(\mathrm{\theta }\right)$ is equal to

• 5

• 3

• 4

• None of these

Domain of the function f(x) = log_x(cos(x)), is

• $\left(- \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right) - \left\{1\right\}$

• $\left[- \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right] - \left\{1\right\}$

• $\left(- \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right)$

• None of these

If x = $\sec \left(\mathrm{\theta }\right) - \cos \left(\mathrm{\theta }\right), \mathrm{y} = {\sec }^{\mathrm{n}}\left(\mathrm{\theta }\right) - {\cos }^{\mathrm{n}}\left(\mathrm{\theta }\right)$, then $\left({\mathrm{x}}^2 + 4\right){\left(\frac{d\mathrm{y}}{d\mathrm{x}}\right)}^2$ is equal to

• n²(y² - 4)

• n²(4 - y²)

• n²(y² + 4)

• None of these

The two curves y = 3 and y = 5 intersect at an angle

• ${\tan }^{-1}\left(\frac{\log \left(3\right) - \log \left(5\right)}{1 + \log \left(3\right)\log \left(5\right)}\right)$

• ${\tan }^{-1}\left(\frac{\log \left(3\right) + \log \left(5\right)}{1 - \log \left(3\right)\log \left(5\right)}\right)$

• ${\tan }^{-1}\left(\frac{\log \left(3\right) + \log \left(5\right)}{1 + \log \left(3\right)\log \left(5\right)}\right)$

• ${\tan }^{-1}\left(\frac{\log \left(3\right) - \log \left(5\right)}{1 - \log \left(3\right)\log \left(5\right)}\right)$

The period of ${\sin }^4\left(\mathrm{x}\right) + {\cos }^4\left(\mathrm{x}\right)$ is

• $\frac{{\mathrm{\pi }}^4}{2}$

• $\frac{{\mathrm{\pi }}^2}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

If 3 cos x $\ne$ 2 sin x, then the general solution of ${\sin }^2\left(\mathrm{x}\right) - \cos \left(2\mathrm{x}\right) = 2 - \sin \left(2\mathrm{x}\right)$ is

• $\mathrm{n\pi } + {\left(- 1\right)}^{\mathrm{n}}\frac{\mathrm{\pi }}{2}, \mathrm{n} \in \mathrm{Z}$

• $\frac{\mathrm{n\pi }}{2}, \mathrm{n} \in \mathrm{Z}$

• $\left(4\mathrm{n} \pm 1\right)\frac{\mathrm{\pi }}{2}, \mathrm{n} \in \mathrm{Z}$

• (2n - 1)$\mathrm{\pi }, \mathrm{n} \in \mathrm{Z}$

If $\cos \left(\mathrm{x}\right) + {\cos }^2\left(\mathrm{x}\right) = 1$, then the value of ${\sin }^{12}\left(\mathrm{x}\right) + 3{\sin }^{10}\left(\mathrm{x}\right) + 3{\sin }^8\left(\mathrm{x}\right) + {\sin }^6\left(\mathrm{x}\right) - 1$, is equal to :

• 2

• 1

• - 1

• 0

The product of all values of $\left(\cos \left(\mathrm{\alpha }\right) + \mathrm{isin}\left(\mathrm{\alpha }\right)\right)$^3/5 is :

• 1

• $\cos \left(\mathrm{\alpha }\right) + \mathrm{isin}\left(\mathrm{\alpha }\right)$

• $\cos \left(3\mathrm{\alpha }\right) + \mathrm{isin}\left(3\mathrm{\alpha }\right)$

• $\cos \left(5\mathrm{\alpha }\right) + \mathrm{isin}\left(5\mathrm{\alpha }\right)$

740.

$\left(1 + \cos \left(\frac{\mathrm{\pi }}{8}\right)\right)\left(1 + \cos \left(\frac{3\mathrm{\pi }}{8}\right)\right)\left(1 + \cos \left(\frac{5\mathrm{\pi }}{8}\right)\right)\left(1 + \cos \left(\frac{7\mathrm{\pi }}{8}\right)\right)$ is equal to

• $\frac{1}{2}$

• $\frac{1}{8}$

• $\cos \left(\frac{\mathrm{\pi }}{8}\right)$

• $\frac{1}{4}$