Question

In ` A B C ,` let `L ,M ,N` be the feet of the altitudes. The prove that `sin(/_M L N)+sin(/_L M N)+sin(/_M N L)=4sinAsinBsinC`

Answer 1

Please refer to video to see the diagram for the given details.
Here,
`/_MLN = 180-2A`
`/_LMN = 180-2B`
`/_MNL= 180-2C`
`:. sin/_MLN+sin/_LMN+sin/_MNL = sin(180-2A)+sin(180-2B)+sin(180-2C)`
`=>sin/_MLN+sin/_LMN+sin/_MNL = sin2A+sin2B+sin2C`
As, in a triangle `ABC, sin2A+sin2B+sin2C = 4sinAsinBsinC`
`:.sin/_MLN+sin/_LMN+sin/_MNL = 4sinAsinBsinC.`