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Question

In $$\displaystyle \Delta ABC $$,let L,M,N be the feet of the altitudes. Then calculate $$\displaystyle \sin(\angle MLN)+\sin(\angle LMN)+\sin(\angle MNL)$$


A
4sinAsinBsinC
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B
3sinAsinBsinC
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C
2sinAsinBsinC
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D
sinAsinBsinC
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Solution

The correct option is A $$\displaystyle 4\sin A\sin B\sin C $$
Using properties of pedal triangle, 
we have $$\displaystyle \angle MLN = 180^\circ -2A $$
$$\displaystyle \angle LMN= 180^\circ -2B $$
$$\displaystyle \angle MLN = 180^\circ -2C $$
$$\displaystyle \Rightarrow \sin(\angle MLN)+\sin(\angle LMN)+\sin(\angle MNL)= \sin 2A+\sin 2B+\sin 2C$$ $$\displaystyle =4\sin A\sin B\sin C $$

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