How do you evaluate the expression \[\cot {180^ \circ }\]?

Hint: We will first convert the given function into its reciprocal trigonometric identity. We will then convert the reciprocal identity in terms of sine and cosine functions. We will then simplify the equation using suitable trigonometric identity and then substitute the value of the angle to get the required value.

Complete step-by-step solution:
Let us write the given expression as below
\[E = \cot {180^ \circ }\]………………………………………\[\left( 1 \right)\]
Now, we know that the cotangent function is equal to the reciprocal of the tangent function. So,
\[\cot x = \dfrac{1}{{\tan x}}\]
Substituting \[x = {180^ \circ }\] in the above equation, we get
\[\cot {180^ \circ } = \dfrac{1}{{\tan {{180}^ \circ }}}\]………………………\[\left( 2 \right)\]
Substituting the equation \[\left( 2 \right)\] in the equation \[\left( 1 \right)\] we get
\[E = \dfrac{1}{{\tan {{180}^ \circ }}}\]……………………………….\[\left( 3 \right)\]
Now, we know that the tangent function is equal to the ratio of the sine function to the cosine function, that is,
\[\tan x = \dfrac{{\sin x}}{{\cos x}}\]……………………………….\[\left( 4 \right)\]
Substituting the equation \[\left( 4 \right)\] in the equation \[\left( 2 \right)\], we get
\[E = \dfrac{{\cos {{180}^ \circ }}}{{\sin {{180}^ \circ }}}\]
Now, the angle \[{180^ \circ }\] can be written as \[{180^ \circ } = {90^ \circ } + {90^ \circ }\]. Writing this in the above equation, we get
\[ \Rightarrow E = \dfrac{{\cos \left( {{{90}^ \circ } + {{90}^ \circ }} \right)}}{{\sin \left( {{{90}^ \circ } + {{90}^ \circ }} \right)}}\]………………………………………..\[\left( 5 \right)\]
Now, we know the trigonometric identities
\[\sin \left( {{{90}^ \circ } + \theta } \right) = \cos \theta \], and
\[\cos \left( {{{90}^ \circ } + \theta } \right) = - \sin \theta \]
So from the above identities, we can write
\[\sin \left( {{{90}^ \circ } + {{90}^ \circ }} \right) = \cos {90^ \circ }\] and \[\cos \left( {{{90}^ \circ } + {{90}^ \circ }} \right) = - \sin {90^ \circ }\]
Substituting these in the equation \[\left( 5 \right)\]., we get
\[ \Rightarrow E = \dfrac{{ - \sin {{90}^ \circ }}}{{\cos {{90}^ \circ }}}\]
Now, we know that \[\sin {90^ \circ } = 1\] and \[\cos {90^ \circ } = 0\].
\[ \Rightarrow E = \dfrac{{ - 1}}{0}\]
As the denominator of the above expression becomes equal to zero, which is not allowed.

Since the above expression is equivalent to the given expression \[\cot {180^ \circ }\], so the expression \[\cot {180^ \circ }\] is undefined.

Note:
The denominator of the given expression is equal to zero. We can make a mistake by concluding that the value of the given expression is equal to infinity. This is a very common misconception in mathematics. We must know that infinity is not a number. It is just an idea that is defined only in the limiting sense. When the denominator of a function approaches zero, we say that the function is approaching infinity. The denominator can only approach zero and can never become zero. So if the denominator of an expression is zero, we say it to be undefined.