I've been trying to figure out why this equation is satisfied: 1 −

Question

I've been trying to figure out why this equation is satisfied:

\frac{1 - (\cos (x))^{3}}{x^{2}} = \frac{2 \cdot (\sin (\frac{x}{2}))^{2}}{x^{2}} \cdot (1 + \cos (x) + (\cos (x))^{2})

but I can't find the proper trigonometric changes in order to change from one to another. I know that the sine comes from the double angle formula, but I obtain slightly different results in other parts and it's never the same as the formula above.

Answer 1

One may observe that, in general,

a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})

giving

1 - (\cos x)^{3} = (1 - \cos x) (1 + \cos x + (\cos x)^{2})

then, by using

2 \sin^{2} (x / 2) = 1 - \cos x

as noticed by the "double-angle" formula

2 \sin^{2} (x / 2) = 1 - \cos x

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