A curious trig identity

Here is an identity that doesn't look correct but it is. For realx andy,

I found the identity in [1]. The author's proof is short. First of all,

Then

Taking square roots completes the proof.

Now note that the statement at the top assumedx andy are real. You can see that this assumption is necessary by, for example, settingx = 2 andy =i.

Where does the proof use the assumption thatx andy are real? Are there weaker assumptions onx andy that are sufficient?

[1] R. M. Robinson. A curious trigonometric identity. American Mathematical Monthly. Vol 64, No 2. (Feb. 1957). pp 83-85

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