Question

Can you explain in detail how to solve this?

Answer 1

`\csc(\arctan(x))=(√(1+x^2))/(x)`

Domain is
`(-\infty,0)\cup(0,\infty)`

Explanation:

Let
`\csc(\arctan(x))=\csc\theta` so that
`\theta=\arctan(x)` and
`\tan\theta=x`:

`\displaystyle \tan\theta=\frac{\text{Opposite}}{\text{Adjacent}}=(x)/(1)\\\\\csc\theta=(1)/(\sin\theta)=\frac{\text{Hypotenuse}}{\text{Opposite}}=(√(1+x^2))/(x)`

You can get these values by drawing a right triangle and labeling each side. Hypotenuse is easily calculated with the Pythagorean Theorem.

Now that we know
`\csc(\arctan(x))=(√(1+x^2))/(x)`, it's clear that
`x\\eq0`, so the domain of the composite trig function is
`(-\infty,0)\cup(0,\infty)`.