# How do you prove [(1+tan^(2)x )/ (tan^(2)x)]=csc^(2)x?

Sep 18, 2016

#### Explanation:

As $1 + {\tan}^{2} x = {\sec}^{2} x$

Hence $\frac{1 + {\tan}^{2} x}{\tan} ^ x = {\sec}^{2} \frac{x}{\tan} ^ x$

= $\frac{\frac{1}{\cos} ^ 2 x}{{\sin}^{2} \frac{x}{\cos} ^ 2 x}$

= $\left(\frac{1}{\cos} ^ 2 x\right) \times \left({\cos}^{2} \frac{x}{\sin} ^ 2 x\right)$

= $\frac{1}{\sin} ^ 2 x$

= ${\csc}^{2} x$