If $A, B, C$ be the angles of a $△ A B C$ ,then the value of $(s i n A + s i n B) (s i n B + s i n C) (s i n C + s i n A)$ is?

< s i n A s i n B s i n C

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> s i n A s i n B s i n C .

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= s i n A s i n B s i n C .

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= c o s A c o s B c o s C

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Solution

The correct option is D $> s i n A s i n B s i n C .$
The sum of any two sides of a triangle is greater than the third side
So,
$a + b > c$ or $b + c > a$ or $a + c > b$

Use the Sine rule, $\frac{s i n A}{a} = \frac{s i n B}{b} = \frac{C}{c}$

$a = k s i n A$ , $b = k s i n B$ and $c = k s i n C$

So,

$k s i n A + k s i n B > K s i n C$

$s i n A + s i n B > s i n C$ Equation $(1)$

Respectively for $b + c > a$ and $c + a > b$

We can say that

$s i n B + s i n C > s i n A$ Equation $(2)$

$s i n A + s i n C > s i n B$ Equation $(3)$

Multiplying all the equation, we get

$(s i n A + s i n B) (s i n B + s i n C) (s i n C + s i n A) > s i n A \times s i n B \times s i n C$

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