Question

If ‘A, B, C’ are acute positive angles, then$\frac{\left[\right(\sin A+\sin B\left)\right(\sin B+\sin C\left)\right(\sin C+\sin A\left)\right]}{\left(\sin A\sin B\sin C\right)}$ is

• A. $>1$
• B. $\le 1$
• C. $=2$
• D. $Noneofthese$

Answer 1

The correct option is D

$Noneofthese$

Explanation for correct option:

Given, ‘A, B, C’ are acute positive angles

Therefore, $\sin A,\sin B\&\sin C$ will be positive.

Also, we know that Arithmetic Mean $\ge$ Geometric Mean

By applying this property on $\sin A,\sin B\&\sin C,$

$$\begin{gathered} \sin A+\sin B\ge 2\sqrt{(\sin A\sin B)}\dots .\left(i\right) \\ \sin B+\sin C\ge 2\sqrt{\left(\sin B\sin C\right)}\dots .\left(ii\right) \\ \sin C+\sin A\ge 2\sqrt{\left(\sin C\sin A\right)}\dots .\left(iii\right) \end{gathered}$$

Multiplying $\left(i\right),\left(ii\right),\left(iii\right)$

$\begin{array}{rcl}(\sin A+\sin B)(\sin B+\sin C)(\sin C+\sin A)& \ge & \left[2\sqrt{\left(\sin A\sin B\right)}\right]\left[2\sqrt{\left(\sin B\sin C\right)}\right]\left[2\sqrt{\left(\sin C\sin A\right)}\right]\\ & \ge & 8\sin A\sin B\sin C\end{array}$

Hence,$\frac{\left[\right(\sin A+\sin B\left)\right(\sin B+\sin C\left)\right(\sin C+\sin A\left)\right]}{\left(\sin A\sin B\sin C\right)}\ge 8$

Hence, correct option is $\mathbf{\left(}\mathit{D}\mathbf{\right)}\mathbf{.}$.