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

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 $b = 1$. So, $(\csc\theta - 1)(\csc\theta+1)=\csc^{2}\theta - 1$.

Step2: Use trigonometric identity

Recall the Pythagorean identity $1+\cot^{2}\theta=\csc^{2}\theta$, which can be rewritten as $\csc^{2}\theta - 1=\cot^{2}\theta$.

Answer:

Since the left - hand side $(\csc\theta - 1)(\csc\theta + 1)$ simplifies to $\csc^{2}\theta - 1$ and $\csc^{2}\theta - 1=\cot^{2}\theta$ (by the Pythagorean identity), the identity $(\csc\theta - 1)(\csc\theta + 1)=\cot^{2}\theta$ is established.