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If A+B+C= pi/2, show that : sin^2 A + si...

If `A+B+C= pi/2`, show that : `sin^2 A + sin^2 B + sin^2 C=1-2 sinA sinB sinC`

Text Solution

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(a) `A+B+C=pi//2` or `2A+2B+2C=pi`
Let `P=2A,Q=2B,R=2CrARrP+Q+R=pi`
`sin^(2)A+sin^(2)B+sin^(2)C`
`=(1-cos2A)/(2)+(1-cos2B)/(2)+(1-cos2C)/(2)`
`=(3-(cos2A+cos2B+cos2C))/(2)`
`=(3-(cosP+cosQ+cosR))/(2)`
`(3-(1+4sin""(P)/(2)sin""(Q)/(2)sin""(P)/(2)))/(2)`
`=1-2sin A sin B sin C`
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