Question

establish the identity (csc θ - 1)(csc θ + 1)= cot² θ. multiply and write the left side expression as the difference of two squares csc² θ - 1. the expression from the previous step is equivalent to cot² θ using what? a. reciprocal identity b. pythagorean identity c. even - odd identity d. quotient identity e. cancellation property

Explanation:

Step1: Expand left - hand side

Using the difference - of - squares formula \((a - b)(a + b)=a^{2}-b^{2}\), where \(a = \csc\theta\) and \(b = 1\), we get \((\csc\theta - 1)(\csc\theta+1)=\csc^{2}\theta - 1\).

Step2: Recall trigonometric identity

One of the Pythagorean identities in trigonometry is \(1+\cot^{2}\theta=\csc^{2}\theta\), which can be rewritten as \(\csc^{2}\theta - 1=\cot^{2}\theta\).

Answer:

B. Pythagorean Identity