Question

Multiply:$\frac{\cos \left(\theta \right)}{1-\sin \left(\theta \right)}\frac{1+\sin \left(\theta \right)}{1+\sin \left(\theta \right)}$

Short answer

On multiplying and simplifying we get,

$sec\left(\theta \right)+\tan \left(\theta \right)$

Step-by-step solution

Given information

We are given an equation$\frac{\cos \left(\theta \right)}{1-\sin \left(\theta \right)}\frac{1+\sin \left(\theta \right)}{1+\sin \left(\theta \right)}$

Multiply 

on multiplying we get,

$$\begin{gathered} \frac{\cos \left(\theta \right)}{1-\sin \left(\theta \right)}\frac{1+\sin \left(\theta \right)}{1+\sin \left(\theta \right)} \\ =\frac{\cos \left(\theta \right)(1+\sin (\theta \left)\right)}{1-{\sin }^2\left(\theta \right)} \\ =\frac{\cos \left(\theta \right)(1+\sin (\theta \left)\right)}{{\cos }^2\left(\theta \right)} \\ =\frac{1+\sin \left(\theta \right)}{\cos \left(\theta \right)} \\ =\frac{1}{\cos \left(\theta \right)}+\frac{\sin \left(\theta \right)}{\cos \left(\theta \right)} \\ =sec\left(\theta \right)+\tan \left(\theta \right) \end{gathered}$$

Conclusion

On multiplying and simplifying we get,

$sec\left(\theta \right)+\tan \left(\theta \right)$