Prove the following identities

#### Solution

\[\text{ LHS }= \frac{\tan^3 x}{1 + \tan^2 x} + \frac{\cot^3 x}{1 + \cot^2 x}\]

\[ = \frac{\tan^3 x}{\sec^2 x} + \frac{\cot^3 x}{{cosec}^2 x}\]

\[ = \frac{\frac{\sin^3 x}{\cos^3 x}}{\frac{1}{\cos^2 x}} + \frac{\frac{\cos^3 x}{\sin^3 x}}{\frac{1}{\sin^2 x}}\]

\[ = \frac{\sin^3 x}{\cos^3 x} \times \frac{\cos^2 x}{1} + \frac{\cos^3 x}{\sin^3 x} \times \frac{\sin^2 x}{1}\]

\[ = \frac{\sin^3 x}{\cos x} + \frac{\cos^3 x}{\sin x}\]

\[ = \frac{\sin^4 x + \cos^4 x}{\sin x \cos x}\]

\[ = \frac{\left( \sin^2 x \right)^2 + \left( \cos^2 x \right)^2}{\sin x \cos x}\]

\[ = \frac{\left( \sin^2 x + \cos^2 x \right)^2 - 2 \sin^2 x \cos^2 x}{\sin x \cos x}\]

\[ = \frac{1^2 - 2 \sin^2 x \cos^2 x}{\sin x \cos x}\]

\[ = \frac{1 - 2 \sin^2 x \cos^2 x}{\sin x \cos x}\]

= RHS

Hence proved.