Question

Given \( (3\cos x - 2\sec x)^2 = 9\cos^2 x + 4\tan^2 x + k \), find \( k \).

• A. 4
• B. 5
• C. 3
• D. 6

Hint:

When solving trigonometric identities, use known relationships such as \( \sec^2 x = 1 + \tan^2 x \) to simplify expressions.

Answer 1

The Correct Option is B

Step 1: Expand the left-hand side.
We start by expanding \( (3\cos x - 2\sec x)^2 \): \[ (3\cos x - 2\sec x)^2 = (3\cos x)^2 - 2(3\cos x)(2\sec x) + (2\sec x)^2 \] \[ = 9\cos^2 x - 12\cos x \sec x + 4\sec^2 x \] Since \( \sec x = \frac{1}{\cos x} \), we can rewrite \( \cos x \sec x = 1 \). Therefore, the expression simplifies to: \[ 9\cos^2 x - 12 + 4\sec^2 x \]
Step 2: Compare with the right-hand side.
The given equation is: \[ 9\cos^2 x + 4\tan^2 x + k \] We know that \( \sec^2 x = 1 + \tan^2 x \), so we can substitute \( 4\sec^2 x = 4(1 + \tan^2 x) \), which simplifies to: \[ 4\sec^2 x = 4 + 4\tan^2 x \] Thus, the left-hand side becomes: \[ 9\cos^2 x - 12 + 4 + 4\tan^2 x = 9\cos^2 x + 4\tan^2 x - 8 \] Now comparing with the right-hand side, we find that: \[ k = 5 \]
Final Answer: 5.