Question

Choose the trigonometric expression below that is equal to cotx + cot^3x to form a trigonometric identity.cot^4xcscx tanxcotx csc^2xsec^2x

Answer 1

Explanation

Background

To solve the exercise, we need to recall two things: the common factor rule, and the cosecant's identity.

The common factor rule says that

$xy+xz=x\cdot(y+z)\text{.}\leftarrow\text{ where x,y, and z are numbers}$

The cosecant's identity says that

$1+\cot ^2(x)=\csc ^2(x).$

Solution of the exercise

Using the equations above, let's play a little with the expression of the exercise:

$\begin{gathered} \cot (x)+\cot ^3(x)=\cot (x)\cdot(1+\cot ^2(x)),\leftarrow\text{ Common factor rule} \\ \cot (x)+\cot ^3(x)=\cot (x)\cdot\csc ^2(x)\text{.}\leftarrow\text{ Cosecant's identity} \end{gathered}$Answer$\cot (x)\cdot\csc ^2(x)\text{.}$