Just double the angle

Calculus Level 3

(j=0(1)jx2j(2j)!)(k=0(1)kx2k+1(2k+1)!)\large \left( \sum_{j=0}^\infty \dfrac{(-1)^{j} x^{2j}}{(2j)!} \right) \left( \sum_{k=0}^\infty \dfrac{(-1)^{k} x^{2k+1}}{(2k+1)!} \right)

Which of the following must be equal to the expression above?

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