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

Ryker Stein 2022-05-19 Answered
I've been trying to figure out why this equation is satisfied:
1 ( cos ( x ) ) 3 x 2 = 2 ( sin ( x 2 ) ) 2 x 2 ( 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.
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Answers (1)

floygdarvn
Answered 2022-05-20 Author has 12 answers
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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