Prove this: sin 5x -2 sin 3x +sinx /cos 5x -cos x = tanx - Maths - Trigonometric Functions

Question

Prove this: sin 5x -2 sin 3x +sinx /cos 5x -cos x = tanx

Abhishek Kumar · Accepted Answer

Dear student, Taking L H S , $\frac{\sin 5 x - 2 \sin 3 x + \sin x}{\cos 5 x - \cos x} = \frac{\sin 5 x + \sin x - 2 \sin 3 x}{\cos 5 x - \cos x}$ $= \frac{2 \sin 3 x \cos 2 x - 2 \sin 3 x}{2 \sin 3 x \sin (- 2 x)} [\begin{matrix} ∵ \cos C - \cos D = 2 \sin (\frac{C + D}{2}) \sin (\frac{D - C}{2}) \\ and \sin C + \sin D = 2 \sin (\frac{C + D}{2}) \cos (\frac{C - D}{2}) \end{matrix}]$ $= \frac{2 \sin 3 x (\cos 2 x - 1)}{- 2 \sin 3 x \sin 2 x} = \frac{1 - \cos 2 x}{\sin 2 x}$ $= \frac{2 \sin^{2} x}{2 \sin x \cos x} [\begin{matrix} ∵ 1 - \cos 2 x = 2 \sin^{2} x \\ and \sin 2 x = 2 \sin x \cos x \end{matrix}]$ ï¿¼=tanx Regards!