Question

Prove that \( \frac{\sec\theta - \tan\theta}{\sec\theta + \tan\theta} = 1 + 2\tan^2\theta - 2\sec\theta\tan\theta \)

Hint:

When you see an expression like \(\frac{A-B}{A+B}\) in a trigonometric proof, a good first step is often to multiply the numerator and denominator by the conjugate, either \(A-B\) or \(A+B\), to simplify the denominator using a Pythagorean identity.

Answer 1

Step 1: Understanding the Concept:
To prove this trigonometric identity, we will simplify the Left Hand Side (LHS) by rationalizing the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator.

Step 2: Key Formula or Approach:
We will use the algebraic identity \((a+b)(a-b) = a^2 - b^2\) and the Pythagorean trigonometric identities: \begin{enumerate} \item \(\sec^2\theta - \tan^2\theta = 1\) \item \(\sec^2\theta = 1 + \tan^2\theta\) \end{enumerate}

Step 3: Detailed Explanation:
Starting with the LHS: \[ \text{LHS} = \frac{\sec\theta - \tan\theta}{\sec\theta + \tan\theta} \] Multiply the numerator and the denominator by the conjugate of the denominator, which is \((\sec\theta - \tan\theta)\): \[ \text{LHS} = \frac{(\sec\theta - \tan\theta) \times (\sec\theta - \tan\theta)}{(\sec\theta + \tan\theta) \times (\sec\theta - \tan\theta)} \] The numerator becomes \((\sec\theta - \tan\theta)^2\) and the denominator becomes \((\sec^2\theta - \tan^2\theta)\). \[ \text{LHS} = \frac{(\sec\theta - \tan\theta)^2}{\sec^2\theta - \tan^2\theta} \] Using the identity \(\sec^2\theta - \tan^2\theta = 1\), the denominator simplifies to 1. \[ \text{LHS} = (\sec\theta - \tan\theta)^2 \] Now, expand the square using \((a-b)^2 = a^2 - 2ab + b^2\): \[ \text{LHS} = \sec^2\theta - 2\sec\theta\tan\theta + \tan^2\theta \] To match the RHS, we need to express everything in terms of \(\tan\theta\). Use the identity \(\sec^2\theta = 1 + \tan^2\theta\): \[ \text{LHS} = (1 + \tan^2\theta) - 2\sec\theta\tan\theta + \tan^2\theta \] Combine the \(\tan^2\theta\) terms: \[ \text{LHS} = 1 + 2\tan^2\theta - 2\sec\theta\tan\theta \] This is equal to the Right Hand Side (RHS).

Step 4: Final Answer:
Since LHS = RHS, the identity is proved.