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If in a triangle acos^2(C/2)+ c cos^2(A...

If in a triangle `acos^2(C/2)+ c cos^2(A/2)=(3b)/2,` then find the relation between the sides of the triangle.

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`a cos^(2).(C)/(2) + cos^(2).(A)/(2) = (3b)/(2)`
or `a (1 + cos C) + c (1 + cos A) = 3b`
or `a+c + (a cos C + c cos A) = 3b`
or `a + c + b = 3b` [bt the projection formula]
or `a + c = 2b`
Hence, a, b, c and in A.P.
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