SOLUTION: Show that sin²x + sin²y + 2sinxsinycos(x+y) = sin²(x+y)

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Question 1139613: Show that sin²x + sin²y + 2sinxsinycos(x+y) = sin²(x+y)
Found 2 solutions by Edwin McCravy, AnlytcPhil:
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Let this be any triangle:



By the law of cosines:

(1)     c%5E2=a%5E2%2Bb%5E2-2%2Aa%2Ab%2Acos%28z%29

By the law of sines:

        a%2Fsin%28x%29=b%2Fsin%28y%29=c%2Fsin%28z%29

Let those equal to k

        a%2Fsin%28x%29=b%2Fsin%28y%29=c%2Fsin%28z%29=k

Then we have 

         

Substitute in (1)

        

        

        

Divide through by k2:

        sin%5E2%28z%29=sin%5E2%28x%29%2Bsin%5E2%28y%29%29-2%2Asin%28x%29%2Asin%28y%29%2Acos%28z%29

Since x + y + z = 180°, z = 180°-x-y = 180°-(x+y)

Use the identities sin%28180%5E%22o%22-theta%29=sin%28theta%29, and cos%28180%5E%22o%22-theta%29=-cos%28theta%29:

          

        

        

Edwin

Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!
Above is your solved problem!

Edwin