If α,β,γ are acute angles and $$cos$$⁡θ$$=$$$$sin$$⁡β$$/$$$$sin$$⁡α,$$cos$$⁡ϕ$$=$$$$sin$$⁡γ$$/$$$$sin$$⁡α and $$cos$$⁡$$($$θ−ϕ$$)=$$$$sin$$⁡β$$sin$$⁡γ then the value of $$tan2$$⁡α−$$tan2$$⁡β−$$tan2$$⁡γ is equal to

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If α,β,γ are acute angles and $$cos$$⁡θ$$=$$$$sin$$⁡β$$/$$$$sin$$⁡α,$$cos$$⁡ϕ$$=$$$$sin$$⁡γ$$/$$$$sin$$⁡α and $$cos$$⁡$$($$θ−ϕ$$)=$$$$sin$$⁡β$$sin$$⁡γ then the value of $$tan2$$⁡α−$$tan2$$⁡β−$$tan2$$⁡γ is equal to

Infinity Learn · Accepted Answer

From the third relation we get  $$cos$$⁡θ$$cos$$⁡ϕ$$+$$$$sin$$⁡θ$$sin$$⁡ϕ$$=$$$$sin$$⁡β$$sin$$⁡γ⇒ $$sin2$$⁡θ$$sin2$$⁡ϕ$$=$$$$cos$$⁡θ$$cos$$⁡ϕ−$$sin$$⁡β$$sin$$⁡γ$$)2$$⇒ $$1$$−$$sin2$$⁡β$$sin2$$⁡α$$1$$−$$sin2$$⁡γ$$sin2$$⁡α$$=$$$$sin$$⁡β$$sin$$⁡γ$$sin2$$⁡α−$$sin$$⁡β$$sin$$⁡γ$$2$$⇒ $$sin2$$⁡α−$$sin2$$⁡β$$sin2$$⁡α−$$sin2$$⁡γ$$=sin2$$⁡β$$sin2$$⁡γ$$1$$−$$sin2$$⁡α$$2$$⇒ $$sin4$$⁡α$$1$$−$$sin2$$⁡β$$sin2$$⁡γ−$$sin2$$⁡α $$sin2$$⁡β$$+sin2$$⁡γ−$$2sin2$$⁡β$$sin2$$⁡γ$$=0$$∴ $$sin2$$⁡α$$=sin2$$⁡β$$+sin2$$⁡γ−$$2sin2$$⁡β$$sin2$$⁡γ$$1$$−$$sin2$$⁡β$$sin2$$⁡γ and $$cos2$$⁡α$$=1$$−$$sin2$$⁡β−$$sin2$$⁡γ$$+sin2$$⁡β$$sin2$$⁡γ$$1$$−$$sin2$$⁡β$$sin2$$⁡γ⇒ $$tan2$$⁡α$$=sin2$$⁡β−$$sin2$$⁡β$$sin2$$⁡γ$$+sin2$$⁡γ−$$sin2$$⁡β$$sin2$$⁡γ$$cos2$$⁡β−$$sin2$$⁡γ$$1$$−$$sin2$$⁡β$$=sin2$$⁡β$$cos2$$⁡γ$$+cos2$$⁡β$$sin2$$⁡γ$$cos2$$⁡β$$cos2$$⁡γ$$=tan2$$⁡β$$+tan2$$⁡γ⇒ $$tan2$$⁡α−$$tan2$$⁡β−$$tan2$$⁡γ$$=0$$

If α,β,γ are acute angles and cos⁡θ=sin⁡β/sin⁡α,cos⁡ϕ=sin⁡γ/sin⁡α and cos⁡(θ−ϕ)=sin⁡βsin⁡γ then the value of tan2⁡α−tan2⁡β−tan2⁡γ is equal to

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