Trigonometric Functions

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

• $\frac{{\mathrm{\pi }}^4}{2}$

• $\frac{{\mathrm{\pi }}^2}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

$\frac{\cos \left(\mathrm{x}\right)}{\cos \left(\mathrm{x} - 2\mathrm{y}\right)} = \mathrm{\lambda } \Rightarrow \tan \left(\mathrm{x} - \mathrm{y}\right)\tan \left(\mathrm{y}\right)$ is equal to

• $\frac{1 + \mathrm{\lambda }}{1 - \mathrm{\lambda }}$

• $\frac{1 - \mathrm{\lambda }}{1 + \mathrm{\lambda }}$

• $\frac{\mathrm{\lambda }}{1 + \mathrm{\lambda }}$

• $\frac{\mathrm{\lambda }}{1 + \mathrm{\lambda }}$

$\cos \left(\mathrm{A}\right)\cos \left(2\mathrm{A}\right)\cos \left(4\mathrm{A}\right) ... \cos \left(2^{\mathrm{n} - 1}\mathrm{A}\right)$ equals

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

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

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

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

If 3cos(x) $\ne$ sin x, then the general solution of sin²(x) - cos(2x) = 2 - sin(2x) is

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

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

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

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

The transformed equation of x² + y² = r² when the axes are rotated through an angle 36° is

• $\sqrt{5}{\mathrm{X}}^2 - 4\mathrm{XY} + {\mathrm{Y}}^2 = {\mathrm{r}}^2$

• ${\mathrm{X}}^2 + 2\mathrm{XY} - \sqrt{5}{\mathrm{Y}}^2 = {\mathrm{r}}^2$

• ${\mathrm{X}}^2 - {\mathrm{Y}}^2 = {\mathrm{r}}^2$

• ${\mathrm{X}}^2 + {\mathrm{Y}}^2 = {\mathrm{r}}^2$

$$\begin{gathered} \mathrm{The} \mathrm{period} \mathrm{of} \left(\tan \left(\mathrm{\theta }\right) - \frac{1}{3}{\tan }^3\left(\mathrm{\theta }\right)\right){\left(\frac{1}{3} - {\tan }^2\left(\mathrm{\theta }\right)\right)}^{ - 1}, \\ \mathrm{where} {\tan }^2\left(\mathrm{\theta }\right) \ne \frac{1}{3} \mathrm{is} \end{gathered}$$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{2\mathrm{\pi }}{3}$

• $\mathrm{\pi }$

• $2\mathrm{\pi }$

$$\begin{gathered} \mathrm{If} {\mathrm{asin}}^2\left(\mathrm{\theta }\right) + {\mathrm{bcos}}^2\left(\mathrm{\theta }\right) = \mathrm{c}, \\ \mathrm{then} {\tan }^2\left(\mathrm{\theta }\right) = ? \end{gathered}$$

• $\frac{\mathrm{b} - \mathrm{c}}{\mathrm{a} - \mathrm{c}}$

• $\frac{\mathrm{c} - \mathrm{b}}{\mathrm{a} - \mathrm{c}}$

• $\frac{\mathrm{a} - \mathrm{c}}{\mathrm{b} - \mathrm{c}}$

• $\frac{\mathrm{a} - \mathrm{c}}{\mathrm{c} - \mathrm{b}}$

If cos(x - y), cos(x), cos(x + y) are three distinct numbers which are in harmoric progression and cos(x) $\ne$ cos(y), then 1 + cos(y) is equal to

• ${\cos }^2\left(\mathrm{x}\right)$

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

• ${\cos }^2\left(\mathrm{x}\right) - 1$

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

$\mathrm{The} \mathrm{set} \mathrm{of} \mathrm{solutions} \mathrm{of} \mathrm{the} \mathrm{equation} \left(\sqrt{3} - 1\right)\sin \left(\mathrm{\theta }\right) + \left(\sqrt{3} + 1\right)\cos \left(\mathrm{\theta }\right) = 2 \mathrm{is}$

• $\left\{2\mathrm{n\pi } \pm \frac{\mathrm{\pi }}{4} + \frac{\mathrm{\pi }}{12} \hspace{0.17em}: \mathrm{n} \in \mathrm{Z}\right\}$

• $\left\{2\mathrm{n\pi } \pm \frac{\mathrm{\pi }}{4} - \frac{\mathrm{\pi }}{12} \hspace{0.17em}: \mathrm{n} \in \mathrm{Z}\right\}$

• $\left\{\mathrm{n\pi } + {\left(- 1\right)}^{\mathrm{n}}\frac{\mathrm{\pi }}{4} + \frac{\mathrm{\pi }}{12} \hspace{0.17em}: \mathrm{n} \in \mathrm{Z}\right\}$

• $\left\{\mathrm{n\pi } + {\left(- 1\right)}^{\mathrm{n}}\frac{\mathrm{\pi }}{4} - \frac{\mathrm{\pi }}{12} \hspace{0.17em}: \mathrm{n} \in \mathrm{Z}\right\}$

$\mathrm{In} \mathrm{a} ∆∆\mathrm{ABC}, \mathrm{C} = 90°. \mathrm{Then}, \frac{{\mathrm{a}}^2 - {\mathrm{b}}^2}{{\mathrm{a}}^2 + {\mathrm{b}}^2} = ?$

• sin(A + B)

• sin(A - B)

• cos(A + B)

• cos(A - B)