K. C. Sinha Solutions for Chapter: Trigonometric Functions (Multiple and Submultiple Angles), Exercise 1: EXERCISE 4.1

If $\tan \theta =\frac{a}{b}$, where $0<\theta <\frac{\pi }{4}$ and $b>a>0$, find the values of $\sin 2\theta ,\cos 2\theta$ and $\tan 2\theta$.

If $\tan \theta =-\frac{3}{4}$ and $\frac{\pi }{2}<\theta <\pi$. Find the values of $\sin \theta ,\cos \theta$ and $\cot \theta$.

If $\frac{3\pi }{4}<\theta <\pi$, prove that $\sqrt{2\cot \theta +\frac{1}{{\sin }^2\theta }}=-1-\cot \theta$.

Prove that $\frac{1+\sin 2\theta }{1-\sin 2\theta }={\left(\frac{1+\tan \theta }{1-\tan \theta }\right)}^2$

Prove that ${\cot }^2\mathrm{A}-{\tan }^2\mathrm{A}=4\cot 2\mathrm{A}·\mathrm{cose}2\mathrm{A}$

Prove that ${\cos }^6\mathrm{A}-{\sin }^6\mathrm{A}=\cos 2\mathrm{A}\left(1-\frac{1}{4}{\sin }^{2}2\mathrm{A}\right)$

If $x\sin \alpha =y\cos \alpha$, prove that $\frac{x}{\sec 2\alpha }+\frac{y}{cosec2a}=x$

If $\alpha =\frac{\pi }{3}$, prove that $\cos \alpha ·\cos 2\alpha ·\cos 3\alpha ·\cos 4\alpha ·\cos 5\alpha \cos 6\alpha =-\frac{1}{16}$