Question:
Simplify
\[
\sec^2 x+5\tan x+5=
\]
Simplify
\[
\sec^2 x+5\tan x+5=
\]
Show Hint
Use the basic identity
\[
\sec^2 x=1+\tan^2 x
\]
to convert expressions involving \(\sec^2 x\) into quadratic expressions in \(\tan x\).
Updated On: Jun 24, 2026
- \((\tan x+2)(\tan x+3)\)
- \((\tan x+1)(\tan x+5)\)
- \((\tan x-2)(\tan x-3)\)
- \((\sin x+2)(\sin x+5)\)
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Use the identity of \(\sec^2 x\).
We know that \[ \sec^2 x=1+\tan^2 x \] Therefore, \[ \sec^2 x+5\tan x+5 = 1+\tan^2 x+5\tan x+5 \] \[ =\tan^2 x+5\tan x+6 \]
Step 2: Factorize the quadratic expression.
Now, \[ \tan^2 x+5\tan x+6 \] can be factorized as \[ \tan^2 x+2\tan x+3\tan x+6 \] \[ =\tan x(\tan x+2)+3(\tan x+2) \] \[ =(\tan x+2)(\tan x+3) \]
Step 3: Final conclusion.
Therefore, \[ \boxed{(\tan x+2)(\tan x+3)} \]
We know that \[ \sec^2 x=1+\tan^2 x \] Therefore, \[ \sec^2 x+5\tan x+5 = 1+\tan^2 x+5\tan x+5 \] \[ =\tan^2 x+5\tan x+6 \]
Step 2: Factorize the quadratic expression.
Now, \[ \tan^2 x+5\tan x+6 \] can be factorized as \[ \tan^2 x+2\tan x+3\tan x+6 \] \[ =\tan x(\tan x+2)+3(\tan x+2) \] \[ =(\tan x+2)(\tan x+3) \]
Step 3: Final conclusion.
Therefore, \[ \boxed{(\tan x+2)(\tan x+3)} \]
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