Question

Establish the identity:$\left(csc\right(\theta )-cot(\theta \left)\right)\left(csc\right(\theta )+cot(\theta \left)\right)=1$

Short answer

We establish the identity$\left(csc\right(\theta )-cot(\theta \left)\right)\left(csc\right(\theta )+cot(\theta \left)\right)=1$

Step-by-step solution

Given information

We are given an expression$\left(csc\right(\theta )-cot(\theta \left)\right)\left(csc\right(\theta )+cot(\theta \left)\right)$

Use algebraic formula (a-b)(a+b)=a2-b2

We get,

$$\begin{gathered} \left(csc\right(\theta )-cot(\theta \left)\right)\left(csc\right(\theta )+cot(\theta \left)\right) \\ =cs{c}^2\left(\theta \right)-co{t}^2\left(\theta \right) \\ Weknowthat1+co{t}^2\left(\theta \right)=cs{c}^2\left(\theta \right) \\ \therefore cs{c}^2\left(\theta \right)-co{t}^2\left(\theta \right)=1 \end{gathered}$$

Conclusion

We establish the identity$\left(csc\right(\theta )-cot(\theta \left)\right)\left(csc\right(\theta )+cot(\theta \left)\right)=1$