Find the sum of the squares of the solutions to <msqrt> 1 &#x2212;<!-- - --> cos

Brenden Tran 2022-06-26 Answered
Find the sum of the squares of the solutions to 1 cos x + 1 + cos x = 3 ,, where π < x < π .
Attempt: Squaring both sides gives
1 cos x + 2 ( 1 cos x ) ( 1 + cos x ) + 1 + cos x = 3 ,
which simplifies to
2 1 cos 2 x = 3 1 sin 2 x = 1 | sin x | = 1.
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Answers (1)

Tianna Deleon
Answered 2022-06-27 Author has 29 answers
Well, when you square the LHS:
( 1 cos ( x ) + 1 + cos ( x ) ) 2 = 2 + 2 1 cos ( x ) 1 + cos ( x ) =
(1) 2 + 2 sin 2 ( x ) = 2 ( 1 + sin 2 ( x ) )
So, we get:
( 1 cos ( x ) + 1 + cos ( x ) ) 2 = 3     2 ( 1 + sin 2 ( x ) ) = 3    
(2) sin 2 ( x ) = 3 2 1 = 1 2 = | sin ( x ) |

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