Complex Analysis: Theory and Applications · Problem 5.4.2

Complex Integration — Howell & Matthews

Mathematics — Complex Analysis · complex-trig-identities

Problem

Demonstrate that, for all zz, sin⁡2z+cos⁡2z=1\sin^2 z + \cos^2 z = 1, as follows. (a) Define the function g(z)=sin⁡2z+cos⁡2zg(z) = \sin^2 z + \cos^2 z. Explain why gg is entire. (b) Show that gg is constant. Hint: Look at g′(z)g'(z). (c) Use part (b) to establish that, for all zz, sin⁡2z+cos⁡2z=1\sin^2 z + \cos^2 z = 1.

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