To find the exact value of [tex]\(\cot \left(\frac{3\pi}{4}\right)\)[/tex], we start by recalling the definition and properties of the cotangent function. Cotangent, written as [tex]\(\cot(\theta)\)[/tex], is the reciprocal of the tangent function:
[tex]\[
\cot(\theta) = \frac{1}{\tan(\theta)}
\][/tex]
Thus, to find [tex]\(\cot \left(\frac{3\pi}{4}\right)\)[/tex], we first need to determine the value of [tex]\(\tan \left(\frac{3\pi}{4}\right)\)[/tex].
The angle [tex]\(\frac{3\pi}{4}\)[/tex] is in the second quadrant of the unit circle. We know that in the second quadrant, the tangent function is negative. Specifically, [tex]\(\frac{3\pi}{4}\)[/tex] relates to the angle [tex]\(\frac{\pi}{4}\)[/tex], where tangent has the value of 1, but taking into account the sign in the second quadrant, we have:
[tex]\[
\tan \left(\frac{3\pi}{4}\right) = -\tan \left(\frac{\pi}{4}\right) = -1
\][/tex]
Now, using the reciprocal relation of cotangent, we find:
[tex]\[
\cot \left(\frac{3\pi}{4}\right) = \frac{1}{\tan \left(\frac{3\pi}{4}\right)} = \frac{1}{-1} = -1
\][/tex]
Thus, the exact value of [tex]\(\cot \left(\frac{3\pi}{4}\right)\)[/tex] is:
[tex]\[
-1
\][/tex]
So, the correct answer is:
[tex]\[
-1
\][/tex]
