Trig Equation

[Thomas154321]

How do I solve

tan^-1 (x) = 2tan^-1 (x-1)?

The answer should be 1.54.

[krtxmrtz]

tan^-1(x) = 2tan^-1 (x-1)

Apply tan to both sides:

x = tan[2tan^-1 (x-1)]

For simplicity, call y = tan^-1 (x-1)

so that tan(y) = (x - 1) and x = tan(2y)

Now use the well known formula:

tan(2x) = 2 tan(x) / (1 - tan²x)

x = tan(2y) = 2tan(y) / (1 - tan²y) = 2(x - 1) / [1 - (x - 1)²]

... and the rest should be easy.

[krtxmrtz]

... and the rest should be easy.

Well, maybe not that easy, as you come up with a third degree equation. I had to use numerical methods (e.g. the Newton-Raphson method) and got approx.
x = 1.544
which corresponds to an angle of 88.45 deg.

[Thomas154321]

Oo I forgot about the double angle forumla Thanks.