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Question 17

Simplify [tex]\(\frac{\sin x}{\sec x + 1}\)[/tex]:

A. [tex]\(\cot x - \cot x \cos x\)[/tex]
B. [tex]\(\cot x + \cot x \cos x\)[/tex]
C. [tex]\(\cot x - \cot x \sin x\)[/tex]
D. [tex]\(\sin x \cos x + \sin x\)[/tex]


Sagot :

To simplify the expression [tex]\(\frac{\sin x}{\sec x + 1}\)[/tex], we need to recall some trigonometric identities and perform the simplification step-by-step. Let's start by analyzing the given expression.

1. Start with the given expression:

[tex]\[ \frac{\sin x}{\sec x + 1} \][/tex]

2. Recall the definition of [tex]\(\sec x\)[/tex]:

[tex]\(\sec x = \frac{1}{\cos x}\)[/tex]

3. Substitute this identity into the expression:

[tex]\[ \frac{\sin x}{\frac{1}{\cos x} + 1} \][/tex]

4. Find a common denominator in the denominator of our fraction:

[tex]\[ \frac{\sin x}{\frac{1 + \cos x}{\cos x}} \][/tex]

5. Simplify the complex fraction by multiplying the numerator and the denominator by [tex]\(\cos x\)[/tex]:

[tex]\[ \frac{\sin x \cdot \cos x}{1 + \cos x} \][/tex]

6. Recognize that this is the simplified form.

Therefore, the simplified expression is:

[tex]\[ \boxed{\frac{\sin x \cos x}{1 + \cos x}} \][/tex]
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