Verify the identity:

Question

arctanx + arctan(1/x)=∏/2

using calculus theory.

(Hint: Differentiate the left hand side of the identity)

Answer 1

Actually, since tan(pi/2-x) = cot(x)
and cot = 1/tan it falls right out.

Or, using the sum of tangents formula,

tan(arctanx + arctan(1/x))
= [tan(arctan(x)) + tan(arctan(1/x))][1 - tan(arctan(x))*tan(arctan(1/x))]

= [x + 1/x]/[1 - x*1/x] = (x + 1/x)/0 = oo

tan pi/2 = oo