How do you simplify \[{\cot ^3}x + {\cot ^2}x + \cot x + 1\]?

Hint: Trigonometry is the branch of mathematics that deals with the relationship between the sides and the angles of the triangles. We have to simplify the given trigonometric equation. To simplify the given trigonometric expression, we will first pair the terms and make groups and then we will use the formula to simplify the terms further.

Complete step-by-step solution:
Given expression:
\[{\cot ^3}x + {\cot ^2}x + \cot x + 1\]
Now we will pair the terms and make the group. So, we have;
\[ \Rightarrow \left( {{{\cot }^3}x + {{\cot }^2}x} \right) + \left( {\cot x + 1} \right)\]
Now we will take \[{\cot ^2}x\] common from the first group. So, we have;
\[ \Rightarrow {\cot ^2}x\left( {\cot x + 1} \right) + \left( {\cot x + 1} \right)\]
Now taking \[\left( {\cot x + 1} \right)\] as common we have;
\[ \Rightarrow \left( {\cot x + 1} \right)\left( {{{\cot }^2}x + 1} \right)\]
Now using the formula that; \[\left( {{{\cot }^2}x + 1} \right) = \cos e{c^2}x\], we have;
\[ \Rightarrow \left( {\cot x + 1} \right)\cos e{c^2}x\]

Note: We can also solve the question by grouping the terms in a different way.
We have;
\[{\cot ^3}x + {\cot ^2}x + \cot x + 1\]
Making the group we get;
\[\left( {{{\cot }^3}x + \cot x} \right) + \left( {{{\cot }^2}x + 1} \right)\]
Taking common we get;
\[ \Rightarrow \cot x\left( {{{\cot }^2}x + 1} \right) + \left( {{{\cot }^2}x + 1} \right)\]
Writing \[\left( {{{\cot }^2}x + 1} \right) = \cos e{c^2}x\], we get;
\[ \Rightarrow \cot x\cos e{c^2}x + \cos e{c^2}x\]
Further taking common we get;
\[ \Rightarrow \cos e{c^2}x\left( {\cot x + 1} \right)\]