prove 2 sin 4 </msup> &#x2061;<!-- ⁡ --> ( x ) + 2 cos 4 </

Carley Haley

Carley Haley

Answered question

2022-04-10

prove 2 sin 4 ( x ) + 2 cos 4 ( x ) + sin 2 ( 2 x ) = 2
2 sin 4 ( x ) + 2 sin 2 ( x ) cos 2 ( x ) + 2 cos 4 ( x ) = 2 ( sin 4 ( x ) + sin 2 ( x ) cos 2 ( x ) + cos 4 ( x ) ) = 2 ( sin 2 ( x ) + cos 2 ( x ) ) 2 sin 2 ( x ) cos 2 ( x ) = 2 ( 1 ) sin 2 ( x ) cos 2 ( x ) = 2 sin 2 ( x ) cos 2 ( x )
And I'm stuck

Answer & Explanation

Erika Ayers

Erika Ayers

Beginner2022-04-11Added 12 answers

sin 2 2 x = ( sin 2 x ) 2 = ( 2 sin x cos x ) 2 = 4 sin 2 x cos 2 x ,
therefore, the LHS is
2 sin 4 x + 4 sin 2 x cos 2 x + 2 cos 4 x ,
which immediately factors as
2 ( sin 2 x + cos 2 x ) 2 = 2.
Berghofaei0e

Berghofaei0e

Beginner2022-04-12Added 3 answers

Using the formula sin ( 2 x ) = 2 sin x cos x we can expand/reduce
2 sin 4 x + 2 cos 4 x + sin 2 ( 2 x ) = 2 sin 4 x + 2 cos 2 x + 4 sin 2 x cos 2 x = 2 ( sin 2 x + cos 2 x ) 2 = 2 ( 1 ) 2

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