Let's solve the given expression [tex]\(\cot x + \tan x\)[/tex] step-by-step.
1. Understanding [tex]\(\cot x\)[/tex] and [tex]\(\tan x\)[/tex]:
- [tex]\(\cot x\)[/tex] is the cotangent of [tex]\(x\)[/tex], which is [tex]\(\frac{\cos x}{\sin x}\)[/tex].
- [tex]\(\tan x\)[/tex] is the tangent of [tex]\(x\)[/tex], which is [tex]\(\frac{\sin x}{\cos x}\)[/tex].
2. Expressing [tex]\(\cot x + \tan x\)[/tex] in terms of [tex]\(\sin x\)[/tex] and [tex]\(\cos x\)[/tex]:
[tex]\[
\cot x + \tan x = \frac{\cos x}{\sin x} + \frac{\sin x}{\cos x}
\][/tex]
3. Finding a common denominator:
- The common denominator for [tex]\(\frac{\cos x}{\sin x}\)[/tex] and [tex]\(\frac{\sin x}{\cos x}\)[/tex] is [tex]\(\sin x \cos x\)[/tex].
- Rewrite the fractions with the common denominator:
[tex]\[
\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} = \frac{\cos^2 x}{\sin x \cos x} + \frac{\sin^2 x}{\sin x \cos x}
\][/tex]
4. Combining the fractions:
- Combine the numerators over the common denominator:
[tex]\[
\frac{\cos^2 x + \sin^2 x}{\sin x \cos x}
\][/tex]
5. Applying the Pythagorean identity:
- Recall the Pythagorean identity: [tex]\(\cos^2 x + \sin^2 x = 1\)[/tex].
- Substitute [tex]\(1\)[/tex] for [tex]\(\cos^2 x + \sin^2 x\)[/tex]:
[tex]\[
\frac{1}{\sin x \cos x}
\][/tex]
Therefore, [tex]\(\cot x + \tan x = \frac{1}{\sin x \cos x}\)[/tex].
The correct answer is (a) [tex]\(\frac{1}{\sin x \cos x}\)[/tex].
