Prove the following identity: `(tan^3theta)/(1+tan^2theta)+(cot^3theta)/(1+cot^2theta)=(1-2sin^2thetacos^2theta)/(sintheta\ costheta)`

Explanation

`\begin{aligned} & \frac{\tan ^{3} \theta}{1+\tan ^{2} \theta}+\frac{\cot ^{3} \theta}{1+\cot ^{2} \theta} \\=& \frac{\tan ^{3} \theta}{\sec ^{2} \theta}+\frac{\cot ^{3} \theta}{\operatorname{cosec}^{2} \theta} \\=& \frac{\frac{\sin ^{3} \theta}{\cos ^{3} \theta}+\frac{\frac{\cos ^{3} \theta}{\sin ^{3} \theta}}{\cos ^{2} \theta}}{\frac{1}{\sin ^{2} \theta}} \\=& \frac{\sin ^{3} \theta}{\cos ^{3}}+\frac{\cos ^{3} \theta}{\sin A} \\=& \frac{\sin ^{4} \theta+\cos ^{4} \theta}{\cos ^{4} \theta \sin \theta} \\=& \frac{\left(\sin ^{2} \theta+\cos ^{2} \theta\right)^{2}-2 \sin ^{2} \theta \cos ^{2} \theta}{\sin \theta \cos \theta} \\=& \frac{1-2 \sin 20 \cos ^{2} \theta}{\sin \theta \cos \theta} \\=& \text { R.H.S } \end{aligned}`