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

Calculate or simplify the expression \( (\csc \theta+\cot \theta)(\csc \theta-\cot \theta) \). Simplify the expression by following steps: - step0: Solution: \(\left(\csc\left(\theta \right)+\cot\left(\theta \right)\right)\left(\csc\left(\theta \right)-\cot\left(\theta \right)\right)\) - step1: Transform the expression: \(\frac{1+\cot\left(\theta \right)\sin\left(\theta \right)}{\sin\left(\theta \right)}\times \left(\csc\left(\theta \right)-\cot\left(\theta \right)\right)\) - step2: Transform the expression: \(\frac{1+\cot\left(\theta \right)\sin\left(\theta \right)}{\sin\left(\theta \right)}\times \frac{1-\cot\left(\theta \right)\sin\left(\theta \right)}{\sin\left(\theta \right)}\) - step3: Multiply the terms: \(\frac{\left(1+\cot\left(\theta \right)\sin\left(\theta \right)\right)\left(1-\cot\left(\theta \right)\sin\left(\theta \right)\right)}{\sin\left(\theta \right)\sin\left(\theta \right)}\) - step4: Multiply the terms: \(\frac{\left(1+\cot\left(\theta \right)\sin\left(\theta \right)\right)\left(1-\cot\left(\theta \right)\sin\left(\theta \right)\right)}{\sin^{2}\left(\theta \right)}\) - step5: Transform the expression: \(\frac{\left(1+\cos\left(\theta \right)\right)\left(1-\cos\left(\theta \right)\right)}{\sin^{2}\left(\theta \right)}\) - step6: Transform the expression: \(\frac{-\left(-\sin^{2}\left(\theta \right)\right)}{\sin^{2}\left(\theta \right)}\) - step7: Calculate: \(\frac{\sin^{2}\left(\theta \right)}{\sin^{2}\left(\theta \right)}\) - step8: Reduce the fraction: \(1\) The identity \((\csc \theta + \cot \theta)(\csc \theta - \cot \theta) = 1\) is established. Now, let's simplify the left side expression by multiplying it: \((\csc \theta + \cot \theta)(\csc \theta - \cot \theta) = \csc^2 \theta - \cot^2 \theta\) Therefore, the identity can be written as: \(\csc^2 \theta - \cot^2 \theta = 1\)