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sina/2 + sinb/2 + sinc/2 = 1 + 4 sin(b+c/4) sin (c + a/4) sin (a+b/4)

$L . H . S . = \sin (\frac{a}{2}) + \sin (\frac{b}{2}) + \sin (\frac{c}{2}) = [\sin (\frac{a}{2}) + \sin (\frac{b}{2})] + \sin (\frac{c}{2}) = [2 \sin (\frac{\frac{a}{2} + \frac{b}{2}}{2}) \cos (\frac{\frac{a}{2} - \frac{b}{2}}{2})] + \cos (\frac{π}{2} - \frac{c}{2}) (\sin A + \sin B = 2 \sin (\frac{A + B}{2}) \cos (\frac{A - B}{2})) = 2 \sin (\frac{a + b}{4}) \cos (\frac{a - b}{4}) + \cos (\frac{π}{2} - \frac{c}{2}) = 2 \sin (\frac{π - c}{4}) \cos (\frac{a - b}{4}) + (1 - 2 \sin^{2} (\frac{π - c}{4})) = 1 + 2 \sin (\frac{π - c}{4}) [\cos (\frac{a - b}{4}) - \sin (\frac{π - c}{4})] = 1 + 2 \sin (\frac{π - c}{4}) [\cos (\frac{a - b}{4}) - \sin (\frac{a + b}{4})] (a + b + c = π \Rightarrow a + b = π - c) = 1 + 2 \sin (\frac{π - c}{4}) [\cos (\frac{a - b}{4}) - \cos (\frac{π}{2} - \frac{a + b}{4})] = 1 + 2 \sin (\frac{π - c}{4}) [2 \sin (\frac{π - b}{4}) \sin (\frac{π - a}{4})] (\cos A - \cos B = 2 \sin (\frac{A + B}{2}) \sin (\frac{B - A}{2})) = 1 + 4 \sin (\frac{π - c}{4}) \sin (\frac{π - b}{4}) \sin (\frac{π - a}{4}) = 1 + 4 \sin (\frac{a + b}{4}) \sin (\frac{a + c}{4}) \sin (\frac{b + c}{4}) = R . H . S . h e n c e p r o v e d .$

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