R S Aggarwal and Veena Aggarwal Solutions for Exercise 1: Exercise 12

Express each of the following in terms of trigonometric ratios of angles lying between $0°$ and $45°$.

$\mathrm{cosec}54°+\sin 72°$

If $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are the angle of a $∆\mathrm{ABC}$, prove that 

$\tan \frac{\mathrm{C}+\mathrm{A}}{2}=\cot \frac{\mathrm{B}}{2}$

If $\cos 2\theta =\sin 4\theta$, where $2\theta$ and $4\theta$ are acute angles, then find the value of $\theta$.

If $\sec 2A=\mathrm{cosec}\left(A-42°\right)$, where $2A$ is an acute angle, then find the value of $A$.

If $\sin 3A=\cos \left(A-26°\right)$, where $3A$ is an acute angle, find the value of $A$.

If $\tan 2A=\cot \left(A-12°\right)$ where $2A$ is an acute angle, find the value of $A$.

If $\sec 4A=\mathrm{cosec}\left(A-15°\right)$ where $4A$ is an acute angle, find the value of $A$.

Prove that:

$\frac{2}{3}{\mathrm{cosec}}^{2}58°-\frac{2}{3}\cot 58°\tan 32°-\frac{5}{3}\tan 13°\tan 37°\tan 45°\tan 53°\tan 77°=-1$