Answer:
Proved
Explanation:
Given
[tex]\frac{(1+tan^2\theta). cot\theta}{cosec^2 \theta}= tan \theta[/tex]
Required
Prove
In trigonometry:
[tex]1 + tan^2\theta = sec^2\theta[/tex]
So, we have:
[tex]\frac{sec^2\theta * cot\theta}{cosec^2 \theta}= tan \theta[/tex]
Express each identity as an inverse:
[tex]\frac{\frac{1}{cos^2\theta} * \frac{1}{tan\theta}}{\frac{1}{sin^2 \theta}}= tan \theta[/tex]
Rewrite as:
[tex](\frac{1}{cos^2\theta} * \frac{1}{tan\theta})/ \frac{1}{sin^2 \theta}= tan \theta[/tex]
Express tan as sin/cos:
[tex](\frac{1}{cos^2\theta} * \frac{1}{sin\theta/cos\theta})/ \frac{1}{sin^2 \theta}= tan \theta[/tex]
[tex](\frac{1}{cos^2\theta} * \frac{1}{sin\theta/cos\theta}) * sin^2 \theta= tan \theta[/tex]
[tex](\frac{1}{cos^2\theta} * \frac{cos\theta}{sin\theta}) * sin^2 \theta= tan \theta[/tex]
[tex](\frac{1}{cos\theta} * \frac{1}{sin\theta}) * sin^2 \theta= tan \theta[/tex]
[tex]\frac{sin^2 \theta}{cos\theta*sin\theta}= tan \theta[/tex]
[tex]\frac{sin \theta}{cos\theta}= tan \theta[/tex]
[tex]tan \theta= tan \theta[/tex]
Proved
