How to prove that \(\displaystyle{{\cos}^{{2}}{\left({z}\right)}}+{{\sin}^{{2}}{\left({z}\right)}}={1}\), where z

Gustavo Lam

Gustavo Lam

Answered question

2022-04-07

How to prove that cos2(z)+sin2(z)=1, where z is a complex variable (if it is true)?

Answer & Explanation

chambasos6

chambasos6

Beginner2022-04-08Added 12 answers

cos2z+sin2z=(cosz+isinz)(coszisinz)=
=eizeiz=
=eiziz=e0=1
slaastro132z

slaastro132z

Beginner2022-04-09Added 8 answers

You can use the identity theorem. As they are just sums of exponentials, sin(z) and cos(z) are holomorphic, and on the real axis sin2(x)+cos2(x)=1. As R is a set with an accumulation point (namely any point in R), they agree everywhere.
First answer is a bit simpler, but this is a good principle to keep in mind when trying to show other identities that are true for real numbers.

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