Question

Factor and simplify:$\frac{3{\sin }^2\left(\theta \right)+4\sin \left(\theta \right)+1}{{\sin }^2\left(\theta \right)+2\sin \left(\theta \right)+1}$

Short answer

The simplified expression can be given as

$\frac{3\sin \left(\theta \right)+1}{\sin \left(\theta \right)+1}$

Step-by-step solution

Given information

We are given am expression$\frac{3{\sin }^2\left(\theta \right)+4\sin \left(\theta \right)+1}{{\sin }^2\left(\theta \right)+2\sin \left(\theta \right)+1}$

We factorize and simplify

We get,

$$\begin{gathered} \frac{3{\sin }^2\left(\theta \right)+4\sin \left(\theta \right)+1}{{\sin }^2\left(\theta \right)+2\sin \left(\theta \right)+1} \\ =\frac{3{\sin }^2\left(\theta \right)+3\sin \left(\theta \right)+\sin \left(\theta \right)+1}{(1+\sin {\left(\theta \right))}^2} \\ =\frac{3\sin \left(\theta \right)\left(\sin \right(\theta )+1)+\left(\sin \right(\theta )+1)}{(1+\sin {\left(\theta \right))}^2} \\ =\frac{\left(3\sin \right(\theta )+1)(1+\sin (\theta \left)\right)}{(1+\sin {\left(\theta \right))}^2} \\ =\frac{3\sin \left(\theta \right)+1}{1+\sin \left(\theta \right)} \end{gathered}$$

Conclusion

The simplified expression can be given as

$\frac{3\sin \left(\theta \right)+1}{\sin \left(\theta \right)+1}$.