K. C. Sinha Solutions for Chapter: Trigonometric Functions (Multiple and Submultiple Angles), Exercise 2: EXERCISE 4.2

Prove that $(\cos A-\cos B{)}^2+(\sin A-\sin B{)}^2=4{\sin }^2\left(\frac{A-B}{2}\right)$.

Prove that ${\cos }^2\frac{\pi }{8}+{\cos }^2\frac{3\pi }{8}+{\cos }^2\frac{5\pi }{8}+{\cos }^2\frac{7\pi }{8}=2$

Prove that ${\sin }^4\frac{\pi }{8}+{\sin }^4\frac{3\pi }{8}+{\sin }^4\frac{5\pi }{8}+{\sin }^4\frac{7\pi }{8}=\frac{3}{2}$.

Prove that $\cot 6°\cot 42°\cot 66°\cot 78°=1$

Prove that $\tan 12°\tan 24°\tan 48°\tan 84°=1$

Prove: $\cos 36°·\cos 72°·\cos 108°·\cos 144°=\frac{1}{16}$

If $\sin \alpha +\sin \beta =a$ and $\cos \alpha +\cos \beta =b$, then prove that $\cos (\alpha -\beta )=\frac{a^2+b^2-2}{2}$.

If $\mathrm{\alpha }$ and $\mathrm{\beta }$ are the two different roots of equation $\mathrm{acos\theta }$ $+\mathrm{bsin\theta }=\mathrm{c},$ prove that $\cot (\mathrm{\alpha }+\mathrm{\beta })=\frac{{\mathrm{a}}^2-{\mathrm{b}}^2}{{\mathrm{a}}^2+{\mathrm{b}}^2}$.