Question

Write the following as an algebraic expression in u, u > 0.

Answer 1

Given

`tan\mleft(cos^(-1)\mleft((u)/(3)\mright)\mright)`

Using the following identity

`tan\mleft(cos^(-1)\mleft(x\mright)\mright)=\frac{\sqrt[]{1-x^2}}{x}`

It follows:

`tan(cos^(-1)((u)/(3)))=\frac{\sqrt[]{1-((u)/(3))^2}}{(u)/(3)}`

Simplify the right hand side

`\frac{\sqrt[]{1-((u)/(3))^2}}{(u)/(3)}=\frac{\sqrt[]{1-(u^2)/(9)}}{(u)/(3)}`

This can further be reduce

`\begin{gathered} \frac{\sqrt[]{1-(u^2)/(9)}}{(u)/(3)} \\ =\frac{\sqrt[]{(9-u^2)/(9)}}{(u)/(3)} \\ =\frac{(1)/(3)\sqrt[]{9-u^2}}{(u)/(3)} \\ =\frac{\sqrt[]{9-u^2}}{3*(u)/(3)} \\ =\frac{\sqrt[]{9-u^2}}{u} \end{gathered}`

Therefore the solution is:

`\frac{\sqrt[]{9-u^2}}{u}`