Question

Rewrite over a common denominator:

$\frac{\sin \left(\theta \right)+\cos \left(\theta \right)}{\cos \left(\theta \right)}+\frac{\sin \left(\theta \right)+\cos \left(\theta \right)}{\sin \left(\theta \right)}$

Short answer

on simplifying over common denominator we get,

$\frac{1+2\sin \left(\theta \right)\cos \left(\theta \right)}{\sin \left(\theta \right)\cos \left(\theta \right)}$

Step-by-step solution

Given information

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

On cross multiplying 

We get,

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

Conclusion

On simplifying we get$\frac{1+2\sin \left(\theta \right)\cos \left(\theta \right)}{\sin \left(\theta \right)\cos \left(\theta \right)}$