Question

Factor and simplify:$\frac{{\cos }^2\left(\theta \right)-1}{{\cos }^2\left(\theta \right)-\cos \left(\theta \right)}$

Short answer

on simplifying we get,$1-sec\left(\theta \right)$

Step-by-step solution

Given information

We are given an expression$\frac{{\cos }^2\left(\theta \right)-1}{{\cos }^2\left(\theta \right)-\cos \left(\theta \right)}$

Simplify the expression

We get,

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

Conclusion

On simplifying we get the expression as$1+sec\left(\theta \right)$