Answered

Find [tex]\csc x[/tex] if [tex]\sin x + \cot x \cos x = \sqrt{3}[/tex].

A. 9
B. 3
C. [tex]\frac{\sqrt{3}}{2}[/tex]
D. [tex]\sqrt{3}[/tex]

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



Answer :

To find [tex]\(\csc x\)[/tex] given that [tex]\(\sin x + \cot x \cos x = \sqrt{3}\)[/tex], let's solve it step-by-step:

1. Understand the given equation:
[tex]\(\sin x + \cot x \cos x = \sqrt{3}\)[/tex]

2. Express [tex]\(\cot x\)[/tex] in terms of [tex]\(\sin x\)[/tex] and [tex]\(\cos x\)[/tex]:
Recall that [tex]\(\cot x = \frac{\cos x}{\sin x}\)[/tex]

3. Substitute [tex]\(\cot x\)[/tex] into the equation:
[tex]\(\sin x + \left(\frac{\cos x}{\sin x}\right) \cos x = \sqrt{3}\)[/tex]

4. Simplify the equation:
[tex]\(\sin x + \frac{\cos^2 x}{\sin x} = \sqrt{3}\)[/tex]

5. Combine the terms over the common denominator [tex]\(\sin x\)[/tex]:
[tex]\(\frac{\sin^2 x + \cos^2 x}{\sin x} = \sqrt{3}\)[/tex]

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

7. Substitute this identity into the equation:
[tex]\(\frac{1}{\sin x} = \sqrt{3}\)[/tex]

8. Solve for [tex]\(\sin x\)[/tex]:
[tex]\(\sin x = \frac{1}{\sqrt{3}}\)[/tex]

9. Determine [tex]\(\csc x\)[/tex]:
[tex]\(\csc x = \frac{1}{\sin x}\)[/tex]

Substitute [tex]\(\sin x\)[/tex] into the equation:
[tex]\(\csc x = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3}\)[/tex]

Therefore, the value of [tex]\(\csc x\)[/tex] is [tex]\(\sqrt{3}\)[/tex].

The best answer from the choices provided is:

D. [tex]\(\sqrt{3}\)[/tex]