Question

If \( \sec \theta + \tan \theta = p \), then \( \sec \theta - \tan \theta = \dots \) “‘latex id="n7o3ls"

• A. \(p\)
• B. \(p^2\)
• C. \( \frac{1}{p} \)
• D. \(1\)

Hint:

Whenever you see \( \sec\theta + \tan\theta \), immediately recall that its reciprocal is \( \sec\theta - \tan\theta \). This is one of the fastest identity-based shortcuts in trigonometry.

Answer 1

The Correct Option is C

Concept: The expressions involving secant and tangent are strongly connected through a fundamental identity: \[ \sec^2\theta - \tan^2\theta = 1 \] This identity is extremely important because it represents a difference of squares structure, which can be factorized as: \[ a^2 - b^2 = (a+b)(a-b) \] So we rewrite: \[ \sec^2\theta - \tan^2\theta = (\sec\theta + \tan\theta)(\sec\theta - \tan\theta) \]

Step 1: Start from the identity. We know: \[ \sec^2\theta - \tan^2\theta = 1 \] Now factorize using algebraic identity: \[ (\sec\theta + \tan\theta)(\sec\theta - \tan\theta) = 1 \]

Step 2: Substitute the given condition. We are given: \[ \sec\theta + \tan\theta = p \] Substitute this into the factored identity: \[ (p)(\sec\theta - \tan\theta) = 1 \]

Step 3: Solve step-by-step for the required expression. To isolate \( \sec\theta - \tan\theta \), divide both sides by \(p\): \[ \sec\theta - \tan\theta = \frac{1}{p} \]

Step 4: Interpretation of result. This result also shows an important reciprocal relationship: \[ (\sec\theta + \tan\theta) \cdot (\sec\theta - \tan\theta) = 1 \] So both expressions are multiplicative inverses of each other. Final Answer: \[ \boxed{\frac{1}{p}} \]