Question

Simplify the expression: \[ \sec^2 x + 5 \tan x + 5 \]

• A. \( (\tan x + 2)(\tan x + 3) \)
• B. \( (\tan x + 1)(\tan x + 5) \)
• C. \( (\tan x - 2)(\tan x - 3) \)
• D. \( (\sin x + 2)(\sin x + 5) \)

Hint:

Always use trigonometric identities like \( \sec^2 x = 1 + \tan^2 x \) to simplify mixed expressions involving \( \sec \) and \( \tan \).

Answer 1

The Correct Option is A

We start with the identity: \[ \sec^2 x = 1 + \tan^2 x \] Substitute into the expression: \[ \sec^2 x + 5 \tan x + 5 = (1 + \tan^2 x) + 5\tan x + 5 = \tan^2 x + 5\tan x + 6 \] Now factor the quadratic: \[ \tan^2 x + 5\tan x + 6 = (\tan x + 2)(\tan x + 3) \]