Question

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

Short answer

On multiplying and simplifying we get,

$csc\left(\theta \right)-cot\left(\theta \right)$

Step-by-step solution

Given information

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

On multiplying and simplifying

We get,

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

Conclusion

On simplifying and multiplying we get,

$csc\left(\theta \right)cot\left(\theta \right)$