Question

Simplify: \[ \frac{\cos x}{1+\sin x}+\tan x= \]

• A. \(1\)
• B. \(\cos x+\sin x\)
• C. \(\sin^2 x\)
• D. \(\sec x\)

Hint:

Whenever expressions contain \[ 1+\sin x, \] try rationalization using \[ 1-\sin x \] to simplify the denominator.

Answer 1

The Correct Option is D

Step 1: Rewrite tan x in terms of sine and cosine.
The expression is $\frac{\cos x}{1+\sin x} + \tan x$. Replace $\tan x = \frac{\sin x}{\cos x}$ to put everything in terms of $\sin$ and $\cos$: \[ \frac{\cos x}{1+\sin x} + \frac{\sin x}{\cos x}. \]
Step 2: Find a common denominator.
The LCD of the two fractions is $\cos x(1+\sin x)$. Combine them: \[ \frac{\cos^2 x + \sin x(1+\sin x)}{\cos x(1+\sin x)}. \]
Step 3: Expand the numerator.
$\cos^2 x + \sin x + \sin^2 x = (\cos^2 x + \sin^2 x) + \sin x = 1 + \sin x$. We used the Pythagorean identity $\cos^2 x + \sin^2 x = 1$.
Step 4: Simplify by cancellation.
\[ \frac{1+\sin x}{\cos x(1+\sin x)} = \frac{1}{\cos x}. \] The factor $(1+\sin x)$ cancels from numerator and denominator.
Step 5: Recognise the result as secant.
$\frac{1}{\cos x} = \sec x$.
Step 6: State the final answer.
\[ \boxed{\sec x} \]