To solve the expression [tex]\(\cos \theta \cdot \sqrt{1 + \cot^2 \theta}\)[/tex], let's break it down step by step:
1. Understanding [tex]\(\cot \theta\)[/tex]:
- Recall that [tex]\(\cot \theta\)[/tex] is the cotangent of [tex]\(\theta\)[/tex], which is defined as [tex]\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)[/tex].
2. Expression Inside the Square Root:
- We need to evaluate [tex]\(1 + \cot^2 \theta\)[/tex].
- Substitute [tex]\(\cot \theta\)[/tex] with [tex]\(\frac{\cos \theta}{\sin \theta}\)[/tex]:
[tex]\[
1 + \cot^2 \theta = 1 + \left( \frac{\cos \theta}{\sin \theta} \right)^2
\][/tex]
- Simplify the expression inside the square root:
[tex]\[
1 + \left( \frac{\cos \theta}{\sin \theta} \right)^2 = 1 + \frac{\cos^2 \theta}{\sin^2 \theta}
\][/tex]
- Combine the terms over a common denominator:
[tex]\[
1 + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta}
\][/tex]
- Use the Pythagorean identity [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex]:
[tex]\[
\frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta} = \frac{1}{\sin^2 \theta}
\][/tex]
3. Square Root of the Expression:
- Now, take the square root of [tex]\(\frac{1}{\sin^2 \theta}\)[/tex]:
[tex]\[
\sqrt{1 + \cot^2 \theta} = \sqrt{\frac{1}{\sin^2 \theta}} = \frac{1}{|\sin \theta|}
\][/tex]
- Since [tex]\(\sqrt{\sin^2 \theta} = |\sin \theta|\)[/tex].
4. Combining with [tex]\(\cos \theta\)[/tex]:
- Multiply [tex]\(\cos \theta\)[/tex] by the result from step 3:
[tex]\[
\cos \theta \cdot \sqrt{1 + \cot^2 \theta} = \cos \theta \cdot \frac{1}{|\sin \theta|}
\][/tex]
- Simplify this expression:
[tex]\[
\cos \theta \cdot \frac{1}{|\sin \theta|} = \frac{\cos \theta}{|\sin \theta|}
\][/tex]
Thus, the detailed step-by-step solution gives us:
[tex]\[
\cos \theta \cdot \sqrt{1 + \cot^2 \theta} = \frac{\cos \theta}{|\sin \theta|}
\][/tex]
However, from the provided final result:
[tex]\[
(\sqrt{\cot(\theta)^2 + 1} \cdot \cos(\theta), \sqrt{\sin(\theta)^{-2}} \cdot \cos(\theta))
\][/tex]
We can also present the final simplified form as:
[tex]\[
\sqrt{\sin(\theta)^{-2}} \cdot \cos(\theta) = \frac{\cos(\theta)}{|\sin(\theta)|}
\][/tex]
But we assume [tex]\(\sin(\theta)\)[/tex] is positive in this case for simplification purposes, yielding:
[tex]\[
\frac{\cos(\theta)}{\sin(\theta)}
\][/tex]
This is our final, simplified form for the given trigonometric expression.
