Question

If \(\sec \theta + \tan \theta = 2 + \sqrt{3}\), then \(\sec \theta - \tan \theta\) is:

• A. \(2 - \sqrt{3}\)
• B. \(\frac{1}{2 - \sqrt{3}}\)
• C. \(\frac{1}{\sqrt{3}}\)
• D. \(\frac{2}{\sqrt{3}}\)
• E. \(\frac{2}{2 - \sqrt{3}}\)

Hint:

Always consider using the conjugate to simplify fractions involving square roots to obtain more straightforward expressions.

Answer 1

The Correct Option is A

Given: \[ \sec \theta + \tan \theta = 2 + \sqrt{3} \] Using the identity for the product of secant and tangent sum and difference: \[ (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = \sec^2 \theta - \tan^2 \theta = 1 \] We can solve for \(\sec \theta - \tan \theta\) by using the given sum: \[ \sec \theta - \tan \theta = \frac{1}{\sec \theta + \tan \theta} = \frac{1}{2 + \sqrt{3}} \] To simplify \(\frac{1}{2 + \sqrt{3}}\), multiply the numerator and the denominator by the conjugate of the denominator: \[ \frac{1}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3} \]