To solve the equation [tex]\(\cot B = \sqrt{\frac{x+4}{x-4}}\)[/tex] for [tex]\(x\)[/tex], follow these steps:
1. Square both sides to eliminate the square root:
[tex]\[
\cot^2 B = \left(\sqrt{\frac{x+4}{x-4}}\right)^2
\][/tex]
Simplifying the right side, we get:
[tex]\[
\cot^2 B = \frac{x+4}{x-4}
\][/tex]
2. Isolate the fraction involving [tex]\(x\)[/tex]:
[tex]\[
\cot^2 B = \frac{x+4}{x-4}
\][/tex]
Multiply both sides by [tex]\(x-4\)[/tex] to get rid of the denominator:
[tex]\[
\cot^2 B \cdot (x - 4) = x + 4
\][/tex]
3. Distribute [tex]\(\cot^2 B\)[/tex] on the left side of the equation:
[tex]\[
\cot^2 B \cdot x - 4 \cot^2 B = x + 4
\][/tex]
4. Move all terms involving [tex]\(x\)[/tex] to one side and constants to the other:
[tex]\[
\cot^2 B \cdot x - x = 4 + 4 \cot^2 B
\][/tex]
Factor [tex]\(x\)[/tex] out of the left side:
[tex]\[
x (\cot^2 B - 1) = 4 (1 + \cot^2 B)
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{4 (1 + \cot^2 B)}{\cot^2 B - 1}
\][/tex]
Now, by using the trigonometric identity [tex]\(\cot^2 B + 1 = \csc^2 B\)[/tex], we get:
[tex]\[
1 + \cot^2 B = \csc^2 B
\][/tex]
Substituting this into our equation for [tex]\(x\)[/tex], we get:
[tex]\[
x = \frac{4 \csc^2 B}{\cot^2 B - 1}
\][/tex]
Thus, the solution for [tex]\(x\)[/tex] in terms of [tex]\(B\)[/tex] is:
[tex]\[
x = \frac{4 \csc^2 B}{\cot^2 B - 1}
\][/tex]
This is the value of [tex]\(x\)[/tex] that satisfies the given equation [tex]\(\cot B = \sqrt{\frac{x+4}{x-4}}\)[/tex].
