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Briana Petty

Briana Petty

Answered question

2022-05-28

Minimizing ( sin 2 ( x ) + 1 sin 2 ( x ) ) 2 + ( cos 2 ( x ) + 1 cos 2 ( x ) ) 2

Answer & Explanation

Liberty Gates

Liberty Gates

Beginner2022-05-29Added 11 answers

Knowing the answer it's not that difficult to get it another way. Let y = cos 2 x. We have sin 2 x = 1 y 2 , cos 2 x = 1 + y 2 . Then
( sin 2 x + 1 sin 2 x ) 2 + ( cos 2 x + 1 cos 2 x ) 2 = ( 1 y 2 + 2 1 y ) 2 + ( 1 + y 2 + 2 1 + y ) 2 = = y 6 + 7 y 4 y 2 + 25 2 ( 1 y 2 ) 2 = = 25 2 + y 2 ( y 4 18 y 2 + 49 ) 2 ( 1 y 2 ) 2 = = 25 2 + y 2 ( 1 2 + 8 1 y 2 + 16 ( 1 y 2 ) 2 )
Since y 2 1, the expression in the brackets is strictly positive.

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