Statement I In any triangle ABC
`a cos A+b cos B+c cos C le s.`
Statement II In any triangle ABC
`sin ((A)/(2))sin ((B)/(2))sin ((C)/(2))le 1/8`

Options

A. Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement- 2 is True.

Explanation

Clearly, LHS of the inequality in statement-2 is a symmetric function of angles A, B and C. So, it is maximum
when A = B =C = 60°. The maximum value of LHSis `(1/2)^(3)=1/8`
`thereforesinA/2sinB/2sinC/2le1/8`
So, statement-2 is true.
Now, state a cos A + b cos B + c cos C - s
`=1/2{2acosA+2bcosB+2ccosC-a-b-c}`
`=1/2{2R (sin 2A + sin 2B + sin 2C)-2R (sin A+ sin B + sin C)|`
`= 4R sin A sin B sin C - 4R cosA/2cosB/2cosC/2`
`=4RcosA/2cosB/2cosC/2(8sinA/2sinB/2sinC/2-1)`
`lt0` [Using statement- 2]
`thereforea cos A + b cos B + c cos C lt s`
So, statement-1 is true and statement-2 is a correct explanation for statement-1.