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.

To establish the identity: \[ (\csc \theta + \cot \theta)(\csc \theta - \cot \theta) = 1 \] we can multiply the left side using the difference of squares formula and then simplify using known trigonometric identities. ### Step 1: Apply the Difference of Squares Formula The expression \((a + b)(a - b)\) is equal to \(a^2 - b^2\). Applying this to the given expression: \[ (\csc \theta + \cot \theta)(\csc \theta - \cot \theta) = \csc^2 \theta - \cot^2 \theta \] ### Step 2: Use Trigonometric Identities Recall the Pythagorean identity involving cosecant and cotangent: \[ \csc^2 \theta = 1 + \cot^2 \theta \] Rearranging this identity gives: \[ \csc^2 \theta - \cot^2 \theta = 1 \] ### Step 3: Substitute Back into the Equation Substituting the result from the trigonometric identity back into our expression: \[ \csc^2 \theta - \cot^2 \theta = 1 \] Thus, we have: \[ (\csc \theta + \cot \theta)(\csc \theta - \cot \theta) = 1 \] ### Conclusion The identity is confirmed by expanding the left side as the difference of two squares and simplifying using fundamental trigonometric identities. **Final Identity:** \[ (\csc \theta + \cot \theta)(\csc \theta - \cot \theta) = \csc^2 \theta - \cot^2 \theta = 1 \]