In $Δ A B C, sin A cos B cos C + sin B cos C cos A + sin C cos A cos B$ is equal to

cos A cos B cos C

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sin A sin B sin C

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None of these

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Solution

The correct option is B $sin A sin B sin C$
In $△ A B C, A + B + C = π$

$\Rightarrow A + B = π - C$ ...... $(i)$

Consider,

$sin A cos B cos C + sin B cos C cos A + sin C cos A cos B$

$= cos C (sin A cos B + cos A sin B) + sin C cos A . cos B$

$= cos C . sin (A + B) + sin C . cos A . cos B$

$= cos C . sin (π - C) + sin C . cos A . cos B$ ...... From $(i)$

$= sin C . cos C + sin C . cos A cos B$

$= sin C [cos C + cos A . cos B]$

$= sin C [cos (π - (A + B)) + cos A . cos B]$

$= sin C [cos A . cos B - cos (A + B)]$

$= sin C [cos A . cos B - (cos A . c o s B - s i n A . s i n B)]$

$= sin C [sin A . sin B]$

$= sin A . sin B . sin C$ .

Suggest Corrections

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