Question:
Simplify:
\[
\frac{\cos x}{1+\sin x}+\tan x=
\]
Simplify:
\[
\frac{\cos x}{1+\sin x}+\tan x=
\]
Show Hint
Whenever expressions contain
\[
1+\sin x,
\]
try rationalization using
\[
1-\sin x
\]
to simplify the denominator.
Updated On: Jun 26, 2026
- \(1\)
- \(\cos x+\sin x\)
- \(\sin^2 x\)
- \(\sec x\)
Show Solution
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The Correct Option is D
Solution and Explanation
Step 1: Write \(\tan x\) in sine-cosine form.
Given, \[ \frac{\cos x}{1+\sin x}+\tan x \] Using \[ \tan x=\frac{\sin x}{\cos x}, \] we get \[ = \frac{\cos x}{1+\sin x}+\frac{\sin x}{\cos x} \]
Step 2: Rationalize the first term.
Multiply numerator and denominator of \[ \frac{\cos x}{1+\sin x} \] by \[ 1-\sin x \] Then, \[ = \frac{\cos x(1-\sin x)} {(1+\sin x)(1-\sin x)} +\frac{\sin x}{\cos x} \]
Step 3: Simplify the denominator.
Using \[ (1+\sin x)(1-\sin x)=1-\sin^2 x, \] we get \[ = \frac{\cos x(1-\sin x)} {\cos^2 x} +\frac{\sin x}{\cos x} \] \[ = \frac{1-\sin x}{\cos x} +\frac{\sin x}{\cos x} \]
Step 4: Combine the fractions.
\[ = \frac{1-\sin x+\sin x}{\cos x} \] \[ = \frac{1}{\cos x} \]
Step 5: Convert into secant form.
Since \[ \sec x=\frac{1}{\cos x}, \] therefore, \[ = \sec x \]
Step 6: Match with the options.
The simplified expression is \[ \sec x \]
Step 7: Final conclusion.
Hence, \[ \boxed{\sec x} \]
Given, \[ \frac{\cos x}{1+\sin x}+\tan x \] Using \[ \tan x=\frac{\sin x}{\cos x}, \] we get \[ = \frac{\cos x}{1+\sin x}+\frac{\sin x}{\cos x} \]
Step 2: Rationalize the first term.
Multiply numerator and denominator of \[ \frac{\cos x}{1+\sin x} \] by \[ 1-\sin x \] Then, \[ = \frac{\cos x(1-\sin x)} {(1+\sin x)(1-\sin x)} +\frac{\sin x}{\cos x} \]
Step 3: Simplify the denominator.
Using \[ (1+\sin x)(1-\sin x)=1-\sin^2 x, \] we get \[ = \frac{\cos x(1-\sin x)} {\cos^2 x} +\frac{\sin x}{\cos x} \] \[ = \frac{1-\sin x}{\cos x} +\frac{\sin x}{\cos x} \]
Step 4: Combine the fractions.
\[ = \frac{1-\sin x+\sin x}{\cos x} \] \[ = \frac{1}{\cos x} \]
Step 5: Convert into secant form.
Since \[ \sec x=\frac{1}{\cos x}, \] therefore, \[ = \sec x \]
Step 6: Match with the options.
The simplified expression is \[ \sec x \]
Step 7: Final conclusion.
Hence, \[ \boxed{\sec x} \]
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