Questions

# If origin is the orthocentre of a triangle formed by the options $\left(\mathrm{cos}\alpha ,\mathrm{sin}\alpha ,0\right),\left(\mathrm{cos}\beta ,\mathrm{sin}\beta ,0\right),\left(\mathrm{cos}\gamma ,\mathrm{sin}\gamma ,0\right)$ then $\sum \mathrm{cos}\left(2\alpha -\beta -\gamma \right)=-$

a
0
b
1
c
2
d
3

detailed solution

Correct option is D

OA=OB=OC;G=H=O(0,0,0)Equilateral trianglecosα+cosβ=−cosγ,sinα+sinβ=−sinγ,square and add cos(α−β)=−12cos(β−γ)=cos(γ−α)cos(2α−β−γ)=cos(α−β)−(γ−α)=1

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