R.K. Singh Solutions for Chapter: Trigonometry, Exercise 1: Model Questions

If $\frac{\sin (\mathrm{x}+\mathrm{y})}{\sin (\mathrm{x}-\mathrm{y})}=\frac{\mathrm{a}+\mathrm{b}}{\mathrm{a}-\mathrm{b}}$ then the value of $\frac{\tan \mathrm{x}}{\tan \mathrm{y}}$ is:

If $\frac{\sin \mathrm{\theta }}{1+\cos \mathrm{\theta }}+\frac{1+\cos \mathrm{\theta }}{\sin \mathrm{\theta }}=4$, then the value of $\mathrm{\theta }$ when $0\le \mathrm{\theta }\le \frac{\mathrm{\pi }}{2}$?

If $2 \sin \mathrm{\theta }=\sec \mathrm{\theta }$, then the value of ${\sin }^4 \mathrm{\theta }+{\cos }^4 \mathrm{\theta }$.

If $\mathrm{x}=\mathrm{a} {\cos }^2 \mathrm{\theta } \sin \mathrm{\theta }, \mathrm{y}=\mathrm{a} {\sin }^2 \mathrm{\theta } \cos \mathrm{\theta }$, then which of the following expressions is independent of $\mathrm{\theta }$?
$1.$ $\frac{{\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)}^3}{\left({\mathrm{x}}^2{\mathrm{y}}^2\right)}$
$2.$ $\frac{\sqrt{\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)}}{\sqrt[3]{\mathrm{xy}}}$

If $\tan \mathrm{\theta }=\frac{\mathrm{x}}{\mathrm{y}}$, then the value of $\frac{\mathrm{x} \sin \mathrm{\theta }+\mathrm{y} \cos \mathrm{\theta }}{\mathrm{x} \sin \mathrm{\theta }-\mathrm{y} \cos \mathrm{\theta }}$.

If $2 {\sin }^2\mathrm{\theta }+{\cos }^{2}45°=\tan 45°$ and $0°\le \mathrm{\theta }\le 90°$, then the value of $\sin \mathrm{\theta }$ is:

In equation, $\mathrm{x}=\cos \mathrm{t}, \mathrm{y}=2 \sin \mathrm{t}.\cos \mathrm{t}$ after eliminating $\mathrm{t}$, we get:

The value of $\frac{\sin 2°\sin 4° \sin 6° \dots \dots \sin 88°}{\cos 88° \cos 86° \cos 84° \dots \dots \cos 2°}$ is: