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If alpha,beta, gamma in(0,(pi)/(2)), the...

If `alpha,beta, gamma in(0,(pi)/(2))`, then prove that `(Sin(alpha+beta+gamma))/(sin alpha+sinbeta+singamma)lt1`.

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`sin(alpha+beta+gamma)=sin alpha cos betacos alpha+cos alpha sin beta cos gamma+cos alpha cos betasin gamma-sin alpha sin beta sin gamma`
or `sin(alpha+beta+gamma)-sin alpha-sin beta-sin gamma`
`=sin alpha(cos betacos gamma-1)+sinbeta(cos alpha cos gamma-1)`
`+sin gamma(cos alpha cos beta-1)-sin alpha sin beta sin gamma`
`therefore sin (alpha+beta+gamma)-sin alpha-sin beta-sin gammalt0`
or `sin (alpha+beta+gamma)ltsin alpha+sin beta+sin gamma`
or `(sin(alpha+beta+gamma))/(sin alpha+sin beta+sin gamma)lt1`
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