A mathematical problem (parametric differentiation)?

Yes you are correct.

x=tan t = (sin t)/(cos t)

so cos t = (sin t)/x

So y^2 = (sin^2 t)/x^2 = (1 - cos^2 t)/x^2 = (1 - y)/x^2

Rearranging this you get y = 1/(x^2 +1)

Yup!

1 + tan^2 t = sec^2 t

So y = cos^2 t = 1/sec^2 t = 1/(1 + tan^2 t) = 1/(1 + x^2)

Hope this helps!

PS This is not parametric differentiation - it is eliminating the parameter

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x^2=tant*tant (given)

=sint*sint/cost*cost

sint*sint=cost*cost*x^2

cost*cost+sint*sint=1

(pythagoras)

y +cost*cost*x^2 =1

y +y*x^2 =1

y(1+x^2) =1

>>>>>>>>> y = 1/(1+x^2)

i hope this helps