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

Step-by-step 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{.}`