Sine and cosine sound the same?

Geometry Level 3

In school, Jimmy was told to calculate the value of the series \[ \sin x + \sin^2 x + \sin^3x + \sin^4x + \cdots \] for some known \(x\). But Jimmy misheard, and he instead calculated the value of the series \[ \cos x + \cos^2 x + \cos^3x + \cos^4x + \cdots . \] Surprisingly, Jimmy got the correct answer!

Assuming that he did his work correctly and that his answer was a negative number, what was the answer?

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