x , y & z are real number such that sin &#x2061;<!-- ⁡ --> <mrow

Brice Colon

Brice Colon

Answered question

2022-05-19

x , y & z are real number such that
sin x + sin y + sin z sin ( x + y + z ) = cos x + cos y + cos z cos ( x + y + z ) = 2
find the value
c y c sin x sin y

Answer & Explanation

Conor Frederick

Conor Frederick

Beginner2022-05-20Added 7 answers

I used the Blue's beautiful idea.
Let e i x = a, e i y = b and e i z = c.
Hence, sin x = a 1 a 2 i = a 2 1 2 a i , cos x = a 2 + 1 2 a , sin y = b 2 1 2 b i , sin x = c 2 1 2 c i and cos x = c 2 + 1 2 c .
Thus, c y c sin x = 2 sin ( x + y + z ) gives c y c ( a 2 b c a b ) = 2 ( a 2 b 2 c 2 1 ) and
c y c cos x = 2 cos ( x + y + z ) gives c y c ( a 2 b c + a b ) = 2 ( a 2 b 2 c 2 + 1 ) or ab+ac+bc=2 and a+b+c=2abc.
Thus,
c y c sin x sin y = c y c ( a 2 1 ) ( b 2 1 ) 4 a b = c y c c ( a 2 1 ) ( b 2 1 ) 4 a b c =
= a b c ( a b + a c + b c ) c y c ( a 2 b + a 2 c ) + a + b + c 4 a b c = a b c ( a b + a c + b c ) ( a + b + c ) ( a b + a c + b c ) + 3 a b c + a + b + c 4 a b c = 1 2 1 + 3 4 + 1 2 = 3 4

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