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Circum Radius

In a ABC, i...

Question

In a $△ A B C,$ if $a$ is the arithmetic mean and $b, c$ are two G.M's between any two positive numbers. Then $\frac{{sin}^{3} B + {sin}^{3} C}{sin A sin B sin C}$ is equal to:

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Solution

Let $a = \frac{x + y}{2}$ and $b = x r, c = x r^{2}$
Then $y = x r^{3}$
Now, $\frac{{sin}^{3} B + {sin}^{3} C}{sin A sin B sin C} = \frac{b^{3} + c^{3}}{a b c}$ using sine rule.
$= \frac{x^{3} r^{3} + x^{3} r^{6}}{\frac{x + x r^{3}}{2} . x r . x r^{2}}$
$= \frac{2 r^{3} (1 + r^{3})}{r^{3} (1 + r^{3})}$
$= 2$

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