This is the first time I've submitted a question so please forgive my inexperience.

I want to simplify the following expression Cos[ArcTan[Xt/Rt]] which is equivalent to 1/Sqrt[1 + Xt^2/Rt^2]

The steps I want to use are:

  1. Square the expression to get rid of the radical.
  2. Multiply the numerator and denominator by the quantity Rt^2.
  3. Simplify the denominator.
  4. Produce the final result by taking the square root.

My code is:

cat = Cos[ArcTan[Xt/Rt]]
$Assumptions = {Rt \[Element] Reals, Rt > 0}
num = Numerator[cat] Rt^2
den = (Denominator[cat]^2) Rt^2 // Simplify
catnew = Sqrt[num/den] // Simplify

catnew === cat

I have three specific questions:

  1. Is there a better way to do this? (I'm sure there is!)
  2. How do I check to see if the original expression (cat) and the new one (catnew) are algebraically equivalent?
  3. How can I submit a question that includes the exact symbolic output from Mathematica?


  • 1
    $\begingroup$ With version 13.2.1, Cos[ArcTan[Xt/Rt]] automatically simplifies to the target form. What version are you using? $\endgroup$
    – Bob Hanlon
    Commented Jun 17, 2023 at 23:05
  • $\begingroup$ @BobHanlon I think the OP meant by "is equivalent to" that it "evaluates to". The phrasing is ambiguous though. -- OP, did you mean Numerator[cat]^2 or are leaving it off since you know the numerator? I use your basic approach frequently. On the specific example, Simplify[Cos[ArcTan[Xt/Rt]], {Xt, Rt} \[Element] Reals] is another way. Also Simplify@Sqrt@Together[Cos[ArcTan[Xt/Rt]]^2]. Not sure if you care where the square root is. $\endgroup$
    – Michael E2
    Commented Jun 17, 2023 at 23:26

1 Answer 1


Q1. Whenever working in an irrational field like inverse trigonometics or elliptic functions, try to rationalize in the power field. Here

   cat^2 /. {a_/b_ :> Sqrt[ a^2/b^2]} // Together // Sqrt//PowerExpand


The results may by wrong with respect to signs or by steps in the complex argument by multiples in $\pi$, but yield a standard algebraic form for small positive variables as a guideline to find a simple target form.

What brings us to Q2:

Fast but not quite True in the complex domain:

   Series[Cos[ArcTan[Xt/Rt]] == Rt/Sqrt[Rt^2 + Xt^2], 
          {Rt, 0, 18}] //PowerExpand


There has much work to be done, to get the true algebraic expressions in the other quadrants

         Cos[ArcTan[(2 + 7 I)/Rt]] - Rt/Sqrt[  Rt^2 + (2 + 7 I)^2], 
            {Rt, -6.0 (1 + I), 6.0 (1 + I)}, 
             AxesLabel -> {"Re", "Im"},BaseStyle -> {FontSize -> 7}]

difference complex plot

Q3: Mathematica Input and Output: Copy Cell as Plain text or Input Text by a right click.

Simply indent the paste here by more than 4 characters. Use Shift+Return to break indented lines.

For nice TeX input, right click cell and copy as LaTeX, paste here in a separate line between \$\$ \$\$

  • $\begingroup$ Thanks for your help. I'm just curious about the rule. I see that it replaces a fraction with the square root of the fraction squared. $$\text{PowerExpand}\left[\text{Together}\left[\sqrt{\text{cat}^2}\right]\right]$$ $$\frac{\text{Rt}}{\sqrt{\text{Rt}^2+\text{Xt}^2}}$$ The command above appears to do the same thing. $\endgroup$
    – Pat Lane
    Commented Jun 19, 2023 at 2:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.