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
B
3sinAsinBsinC
C
2sinAsinBsinC
D
sinAsinBsinC

Solution

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

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