Question

if tan(x-iy) = u-iv, show that u^2+v^2+2ucot2x = 1​

Answer 1

We have,tan(x+iy) = u+ivtan(x−iy) = u − ivNow, tan 2x = tan[(x+iy)+(x−iy)]⇒tan 2x = tan(x+iy) + tan(x−iy)1 − tan(x+iy) × tan(x−iy)⇒1cot 2x = u+iv+u−iv1−(u2+v2)⇒1cot 2x = 2u1−u2−v2⇒u2+v2+2u cot 2x = 1