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]
