Prove the following identities: ^3x 1+ ^2x + ^3x 1+ ^2x = 1-2 ^2x…

Question

Prove the following identities: $\frac{\tan^3\text{x}}{1+\tan^2\text{x}}+\frac{\cot^3\text{x}}{1+\cot^2\text{x}}=\frac{1-2\sin^2\text{x}\cos^2\text{x}}{\sin\text{x}\cos\text{x}}$

✓

Answer

$\text{L.H.S}=\frac{\tan^{3}\text{x}}{1+\tan^{2}\text{x}}+\frac{\cot^{3}\text{x}}{1+\cot^{2}\text{x}}$ $=\frac{\sin^{3}\text{x}}{\cos^{3}\text{x}\bigg(1+\frac{\sin^{2}\text{x}}{\cos^{2}\text{x}}\bigg)}+\frac{\cos^{3}\text{x}}{\sin^{3}\text{x}\bigg(1+\frac{\cos^{2}\text{x}}{\sin^{2}\text{x}}\bigg)}$$$ $\Big(\because\tan\text{x}=\frac{\sin\text{x}}{\cos\text{x}}\text{ and }\cot\text{x}=\frac{\cos\text{x}}{\sin\text{x}}\Big)$ $=\frac{\sin^{3}\text{x}\cos^{2}\text{x}}{\cos^{3}\text{x}\big(\cos^{2}\text{x}+\sin^{2}\text{x}\big)}+\frac{\cos^{3}\text{x}\sin^{2}\text{x}}{\sin^{3}\text{x}\big(\sin^{2}\text{x}+\cos^{2}\text{x}\big)}$$$ $=\frac{\sin{3}\text{ x}}{\cos\text{x}}+\frac{\cos^{3}\theta}{\sin\text{x}}$ $\big(\because\cos^{2}\text{x}+\sin^{2}\text{x}=1\big)$ $=\frac{\sin^{4}\text{x}+\cos^{4}\text{x}}{\sin\text{x}\cos\text{x}}$ $=\frac{\big(\sin^{2}\text{x}\big)^{2}+\big(\cos^{2}\text{x}\big)^{2}+2\sin^{2}\text{x}\cos^{2}\text{x}-2\sin^{2}\text{x}\cos^{2}\text{x}}{\sin\text{x}\cos\text{x}}$ $\big(\text{adding and subtracting }2\sin^{2}\text{x}\cos^{2}\text{x}\big)$ $=\frac{\big(\sin^{2}\text{x}+\cos^{2}\text{x}\big)^{2}-2\sin^{2}\text{x}\cos^{2}\text{x}}{\sin\text{x}\cos\text{x}}$ $=\frac{1^{2}-2\sin^{2}\text{x}\cos^{2}\text{x}}{\sin\text{x}\cos\text{x}}$ $\big(\because\sin^{2}\text{x}+\cos^{2}\text{x}=1\big)$ $=\frac{1-2\sin^{2}\text{x}\cos^{2}\text{x}}{\sin\text{x}\cos\text{x}}$ $=\text{R.H.S}$ $\text{Proved}$