Question

I'm doing some revision on PDEs, and i've stumbled upon something curious - i'm wondering if anyone can explain this to me?

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

So why when I integrate each of these expressions in Mathcad does it give me two different results? Are they in fact the same thing, and if so, how?

Mathcad says:

Integral of 1/tan(x) dx = ln tan(x) - 0.5(ln (1+tan^2(x)).

Integral of cos(x)/tan(x) dx = ln sin(x)

Answer 1

tan (x) can be expressed as sin(x) / cos(x) . If you look up the difinition of tan, which is opposite/adjacent, and compare it with the definition of sin(x) / cos (x), which is (op/hyp )/(ad/hyp), youll see that they are the same. And so are the integrals (your first should just be a more difficult representation of your second)

Answer 2

they are equivalent answers just in different forms
ln(tan) = ln(sin) - ln(cos)
ln(cos) + .5(ln(1+tan^2) = 0

Answer 3

I think i get that... thank you both