Establish the identity. \[ (\sec \theta+\tan \theta)(\sec \theta-\boldsymbol{\operatorname { t a n }} \theta)=1 \] Multiply and write the left side expression as the difference of two squares.

To establish the identity \[ (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 1, \] we can multiply the left side and express it as the difference of two squares. **Step-by-Step Explanation:** 1. **Multiply the Binomials:** \[ (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) \] This is in the form \((a + b)(a - b)\), which equals \(a^2 - b^2\). 2. **Apply the Difference of Squares:** \[ (\sec \theta)^2 - (\tan \theta)^2 \] 3. **Use Pythagorean Identity:** Recall the Pythagorean identity: \[ \sec^2 \theta = 1 + \tan^2 \theta \] Rearranging this identity gives: \[ \sec^2 \theta - \tan^2 \theta = 1 \] 4. **Substitute Back:** \[ (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = \sec^2 \theta - \tan^2 \theta = 1 \] **Conclusion:** By expanding the left side as the difference of two squares and applying the Pythagorean identity, we've established that: \[ (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 1 \] Answer: After expanding, the left side is sec²θ – tan²θ, which equals 1. Thus, (sec θ + tan θ)(sec θ – tan θ) = sec²θ − tan²θ = 1