Establish the identity. \[ (\csc \theta-1)(\csc \theta+1)=\cot ^{2} \theta \] Multiply and write the left side expression as the difference of two squares.

Calculate or simplify the expression \( (\csc \theta-1)(\csc \theta+1) \). Simplify the expression by following steps: - step0: Solution: \(\left(\csc\left(\theta \right)-1\right)\left(\csc\left(\theta \right)+1\right)\) - step1: Transform the expression: \(\frac{1-\sin\left(\theta \right)}{\sin\left(\theta \right)}\times \left(\csc\left(\theta \right)+1\right)\) - step2: Transform the expression: \(\frac{1-\sin\left(\theta \right)}{\sin\left(\theta \right)}\times \frac{1+\sin\left(\theta \right)}{\sin\left(\theta \right)}\) - step3: Multiply the terms: \(\frac{\left(1-\sin\left(\theta \right)\right)\left(1+\sin\left(\theta \right)\right)}{\sin\left(\theta \right)\sin\left(\theta \right)}\) - step4: Multiply the terms: \(\frac{\left(1-\sin\left(\theta \right)\right)\left(1+\sin\left(\theta \right)\right)}{\sin^{2}\left(\theta \right)}\) - step5: Rewrite the expression: \(\frac{\cos^{2}\left(\theta \right)}{\sin^{2}\left(\theta \right)}\) - step6: Rewrite the expression: \(\left(\frac{\cos\left(\theta \right)}{\sin\left(\theta \right)}\right)^{2}\) - step7: Transform the expression: \(\cot^{2}\left(\theta \right)\) The identity \((\csc \theta-1)(\csc \theta+1)=\cot ^{2} \theta\) is established by simplifying the left side expression as the difference of two squares, which results in \(\cot ^{2} \theta\).