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In triangle A B C , prove that sinA/2+si...

In triangle `A B C ,` prove that `sinA/2+sinB/2+sinC/2lt=3/2dot` Hence, deduce that `cos(pi+A)/4cos(pi+B)/4cos(pi_C)/4lt=1/8`

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`sin(A)/(2)+sin(B)/(2)+sin(C)/(2)`
`=2sin(A+B)/(4)cos(A-B)/(4)+sin(C)/(2)`
`=2sin(pi-C)/(4)cos(A-B)/(4)+cos(pi-C)/(2)`
`=2sin(pi-C)/(4)cos(A-B)/(4)+1-2sin^(2)(pi-C)/(4)`
`le1+2sin(pi-C)/(4)-2sin^(2)(pi-C)/(4)(because cos(A-B)/(4)le1)`
`=1-2[(sin(pi-C)/(4)-(1)/(2))^(2)-(1)/(4)]`
`=(3)/(2)-2(sin(pi-C)/(4)-(1)/(2))^(2)le(3)/(2)`
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