To find the exact value of [tex]\( f\left(\frac{\pi}{3}\right) \)[/tex] where [tex]\( f(x) = \cot^2 x \)[/tex], we follow these steps:
1. Understand the function and the trigonometric identity:
The cotangent function [tex]\( \cot(x) \)[/tex] is the reciprocal of the tangent function [tex]\( \tan(x) \)[/tex], so:
[tex]\[
\cot(x) = \frac{1}{\tan(x)}
\][/tex]
Therefore, [tex]\( \cot^2(x) = \left(\frac{1}{\tan(x)}\right)^2 = \frac{1}{\tan^2(x)} \)[/tex].
2. Evaluate [tex]\( x \)[/tex] at [tex]\( \frac{\pi}{3} \)[/tex]:
We need to determine [tex]\( \tan\left(\frac{\pi}{3}\right) \)[/tex]. It is known that:
[tex]\[
\tan\left(\frac{\pi}{3}\right) = \sqrt{3}
\][/tex]
3. Square the tangent value:
Since [tex]\( \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \)[/tex]:
[tex]\[
\tan^2\left(\frac{\pi}{3}\right) = (\sqrt{3})^2 = 3
\][/tex]
4. Calculate [tex]\( \cot^2\left(\frac{\pi}{3}\right) \)[/tex]:
Using the relationship [tex]\( \cot^2 x = \frac{1}{\tan^2 x} \)[/tex]:
[tex]\[
\cot^2\left(\frac{\pi}{3}\right) = \frac{1}{\tan^2\left(\frac{\pi}{3}\right)} = \frac{1}{3}
\][/tex]
Thus, the exact value of [tex]\( f\left(\frac{\pi}{3}\right) \)[/tex] in its simplest form is:
[tex]\[
\boxed{\frac{1}{3}}
\][/tex]
