To rewrite the expression [tex]\(\sqrt{4x^2 + 16}\)[/tex] using the given substitution [tex]\(\frac{x}{2} = \cot(\theta)\)[/tex], we follow these steps:
1. Substitute [tex]\(x\)[/tex] in terms of [tex]\(\theta\)[/tex]:
Given [tex]\(\frac{x}{2} = \cot(\theta)\)[/tex], we can solve for [tex]\(x\)[/tex]:
[tex]\[
x = 2 \cot(\theta)
\][/tex]
2. Substitute [tex]\(x\)[/tex] into the original expression:
[tex]\[
\sqrt{4x^2 + 16} = \sqrt{4(2 \cot(\theta))^2 + 16}
\][/tex]
3. Simplify the expression inside the square root:
[tex]\[
\sqrt{4(2 \cot(\theta))^2 + 16} = \sqrt{4 \cdot 4 \cot^2(\theta) + 16} = \sqrt{16 \cot^2(\theta) + 16}
\][/tex]
4. Factor out the common factor inside the square root:
[tex]\[
\sqrt{16 \cot^2(\theta) + 16} = \sqrt{16 (\cot^2(\theta) + 1)}
\][/tex]
5. Take the constant outside the square root:
[tex]\[
\sqrt{16 (\cot^2(\theta) + 1)} = 4 \sqrt{\cot^2(\theta) + 1}
\][/tex]
6. Use the trigonometric identity for simplification:
We know that [tex]\(\cot^2(\theta) + 1 = \csc^2(\theta)\)[/tex] (where [tex]\(\csc(\theta)\)[/tex] is the cosecant function, defined as [tex]\(\csc(\theta) = \frac{1}{\sin(\theta)}\)[/tex]).
[tex]\[
4 \sqrt{\cot^2(\theta) + 1} = 4 \sqrt{\csc^2(\theta)}
\][/tex]
7. Simplify the square root of the square:
[tex]\[
4 \sqrt{\csc^2(\theta)} = 4 \csc(\theta)
\][/tex]
Thus, the simplified trigonometric expression is:
[tex]\[
4 \csc(\theta)
\][/tex]
