Question

Evaluate \( \left[ 1 + \sec 2\theta \right] \left[ 1 + \sec 40^\circ \right] \):

• A. \( \tan \theta \tan 40^\circ \)
• B. \( 4 \cot \theta \tan 40^\circ \)
• C. \( \cot \theta \tan 40^\circ \)
• D. \( 4 \tan \theta \tan 40^\circ \)

Hint:

For trigonometric expressions, use identities such as \( \sec x = \frac{1}{\cos x} \) and simplify step by step.

Answer 1

The Correct Option is C

We are given the expression \( \left[ 1 + \sec 2\theta \right] \left[ 1 + \sec 40^\circ \right] \). Step 1: Use the identity \( \sec x = \frac{1}{\cos x} \) to rewrite the terms: \[ 1 + \sec 2\theta = 1 + \frac{1}{\cos 2\theta}, \quad 1 + \sec 40^\circ = 1 + \frac{1}{\cos 40^\circ} \] Step 2: Multiply the two expressions and simplify: \[ \left(1 + \frac{1}{\cos 2\theta} \right) \left(1 + \frac{1}{\cos 40^\circ} \right) \] After simplifying, we get the final result as \( \cot \theta \tan 40^\circ \). Thus, the correct answer is \( \cot \theta \tan 40^\circ \).