Question

Simplify \[ \sec^2 x+5\tan x+5= \]

• 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:

Use the basic identity \[ \sec^2 x=1+\tan^2 x \] to convert expressions involving \(\sec^2 x\) into quadratic expressions in \(\tan x\).

Answer 1

The Correct Option is A

Step 1: Use the Pythagorean identity to replace $\sec^2 x$.
We know $\sec^2 x = 1 + \tan^2 x$. Substitute: \[ \sec^2 x + 5\tan x + 5 = 1 + \tan^2 x + 5\tan x + 5 = \tan^2 x + 5\tan x + 6 \]

Step 2: Factor the quadratic in $\tan x$.
We need two numbers that multiply to $6$ and add to $5$: those are $2$ and $3$. \[ \tan^2 x + 5\tan x + 6 = (\tan x + 2)(\tan x + 3) \]

Step 3: Verify by expanding.
$(\tan x+2)(\tan x+3) = \tan^2 x+3\tan x+2\tan x+6 = \tan^2 x+5\tan x+6$. Correct.

Step 4: Match with the options.
Option 1 is $(\tan x+2)(\tan x+3)$. This matches exactly.

Step 5: Note why other options fail.
$(\tan x+1)(\tan x+5) = \tan^2 x+6\tan x+5 \neq$ our expression. $(\tan x-2)(\tan x-3) = \tan^2 x-5\tan x+6 \neq$ our expression (wrong signs).

Step 6: State the answer.
\[ \boxed{(\tan x+2)(\tan x+3)} \]