Answered

Given [tex]\frac{1+\cos x}{\sin x}+\frac{\sin x}{1+\cos x}=4[/tex], find a numerical value of one trigonometric function of [tex]x[/tex].

A. [tex]\tan x = 2[/tex]
B. [tex]\sin x = 2[/tex]
C. [tex]\tan x = \frac{1}{2}[/tex]
D. [tex]\sin x = \frac{1}{2}[/tex]

Please select the best answer from the choices provided.
A
B
C
D



Answer :

Alright, let's solve the given equation step by step.

The given equation is:
[tex]\[ \frac{1 + \cos x}{\sin x} + \frac{\sin x}{1 + \cos x} = 4 \][/tex]

First, let's rewrite this equation in a more workable form. Notice that [tex]\(\frac{1 + \cos x}{\sin x}\)[/tex] can be seen as one term and [tex]\(\frac{\sin x}{1 + \cos x}\)[/tex] as another. To find a common denominator, we multiply both sides accordingly.

The next step is to work with a common denominator on the left side:
[tex]\[ \frac{(1 + \cos x)^2 + \sin^2 x}{\sin x (1 + \cos x)} = 4 \][/tex]

Simplify the numerator:
[tex]\[ (1 + \cos x)^2 + \sin^2 x = 1 + 2\cos x + \cos^2 x + \sin^2 x \][/tex]

Using the Pythagorean identity [tex]\(\sin^2 x + \cos^2 x = 1\)[/tex]:
[tex]\[ 1 + 2\cos x + 1 = 2 + 2\cos x \][/tex]

Thus:
[tex]\[ \frac{2 + 2\cos x}{\sin x (1 + \cos x)} = 4 \][/tex]

Now we simplify further:
[tex]\[ \frac{2(1 + \cos x)}{\sin x (1 + \cos x)} = 4 \][/tex]

Here, [tex]\((1 + \cos x)\)[/tex] cancels out from the numerator and the denominator (assuming [tex]\(\cos x \neq -1\)[/tex]):
[tex]\[ \frac{2}{\sin x} = 4 \][/tex]

Thus:
[tex]\[ 2 = 4\sin x \][/tex]

Solving for [tex]\(\sin x\)[/tex]:
[tex]\[ \sin x = \frac{2}{4} = \frac{1}{2} \][/tex]

Therefore, the numerical value of the trigonometric function of [tex]\(x\)[/tex] is [tex]\(\sin x = \frac{1}{2}\)[/tex].

From the given options:
a. [tex]\(\tan x = 2\)[/tex]
b. [tex]\(\sin x = 2\)[/tex]
c. [tex]\(\tan x = \frac{1}{2}\)[/tex]
d. [tex]\(\sin x = \frac{1}{2}\)[/tex]

The correct answer is:
d. [tex]\(\sin x = \frac{1}{2}\)[/tex]