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If the sides a , ba n dCof A B C are in ...

If the sides `a , ba n dCof A B C` are in `AdotPdot,` prove that `2sinA/2sinC/2=sinB/2` `acos^2C/2+cos^2A/2=(3b)/2`

Text Solution

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(i) `2 sin.(A)/(2) sin.(C)/(2) = 2 sqrt(((s-b) (s-c))/(bc)). sqrt(((s -a) (s-b))/(ab))`
`= (2(s-b))/(b) sqrt(((s-a) (s-c))/(ac))`
`= (2s - 2b)/(b). sin.(B)/(2)`
`=(a + c - b)/(b). sin.(B)/(2)`
`= (2b -b)/(b). sin.(B)/(2)` (as a, b, c are in A.P.)
`= sin.(B)/(2)`
(ii) `a cos^(2).(C)/(2) + c cos^(2).(A)/(2) = (a s (s - c))/(ab) + (c s(s -a))/(bc)`
`= (s)/(b) (s - c + s -a)`
`= (a + b+ c)/(2b) b`
`= (2b + b)/(2)`
`= (3b)/(2)`
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