Let $$tan$$α,$$tan$$β,$$tan$$γ ;α,β,γ,γ≠(2n-1)π$$2$$,n∈N be the slopes of three line segments OA,OB and OC respectively, where O is origin. If Circum center of ΔABC coincides with origin and its ortho center lies on $$y-axis$$, the value of $$cos3$$α$$+cos3$$β$$+cos3$$γ$$cos$$α$$cos$$β$$cos$$γ$$2$$ is equal to

Question

Let $$tan$$α,$$tan$$β,$$tan$$γ ;α,β,γ,γ≠(2n-1)π$$2$$,n∈N be the slopes of three line segments OA,OB and  OC respectively, where O is origin. If Circum center of ΔABC coincides with origin and  its ortho center lies on $$y-axis$$, the value of $$cos3$$α$$+cos3$$β$$+cos3$$γ$$cos$$α$$cos$$β$$cos$$γ$$2$$ is equal to

Infinity Learn · Accepted Answer

$$A=($$$$cos$$α,$$sin$$α$$)$$,$$B=($$$$cos$$β,$$sin$$β$$)$$,$$C=($$$$cos$$γ,$$sin$$γ$$)$$ Circum center $$=(0$$,$$0)Centroid$$ G divides the HS in the ratio $$2$$:$$1$$ internally   Ortho center $$H=($$$$cos$$α$$+$$$$cos$$β$$+$$$$cos$$γ,$$sin$$α$$+$$$$sin$$β$$+$$$$sin$$γ$$)$$ lies on y $$-axis$$ ⇒$$cos$$α$$+$$$$cos$$β$$+$$$$cos$$γ$$=0$$⇒$$cos3$$α$$+cos3$$β$$+cos3$$γ$$=3cos$$α$$cos$$β$$cos$$γ∑$$cos3$$α$$=4$$∑$$cos3$$α$$-3$$∑$$cos$$α$$=12cos$$α$$cos$$β$$cos$$γ$$+0$$ Given expression $$=12cos$$α$$cos$$β$$cos$$γ$$cos$$α$$cos$$β$$cos$$γ$$2=144$$

Let tanα,tanβ,tanγ ;α,β,γ,γ≠(2n-1)π2,n∈N be the slopes of three line segments OA,OB and OC respectively, where O is origin. If Circum center of ΔABC coincides with origin and its ortho center lies on y-axis, the value of cos3α+cos3β+cos3γcosαcosβcosγ2 is equal to

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