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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