Question

If \[ \sec x + \tan x = 3, \quad x \in \left( 0, \frac{\pi}{2} \right), \] then \( \sin x = \)

• A. \( \frac{3}{5} \)
• B. \( \frac{4}{5} \)
• C. -1
• D. \( \frac{1}{5} \)

Hint:

When given \( \sec x + \tan x \), square the equation and use trigonometric identities to simplify and find the value of \( \sin x \).

Answer 1

The Correct Option is B

Step 1: Express \( \sec x \) and \( \tan x \).
We are given \( \sec x + \tan x = 3 \). Using the identity \( \sec^2 x - \tan^2 x = 1 \), we square both sides: \[ (\sec x + \tan x)^2 = 9. \] Expanding the left-hand side: \[ \sec^2 x + 2 \sec x \tan x + \tan^2 x = 9. \] Using the identity \( \sec^2 x = 1 + \tan^2 x \), we substitute and simplify to find \( \sec x \).
Step 2: Solve for \( \sin x \).
After finding \( \sec x \), we use \( \sin^2 x = 1 - \cos^2 x \) to compute \( \sin x \), yielding \( \sin x = \frac{4}{5} \). Thus, the correct answer is option (B).