If α,β,γ,δ are the solutions of the equation $$tan$$⁡θ$$+$$π$$4=$$ $$3$$ $$tan3$$θ, no two of which have equal tangents.The value of $$tan$$⁡α$$+$$$$tan$$⁡β$$+$$$$tan$$⁡γ$$+$$$$tan$$⁡δ isThe value of $$tan$$⁡α $$tan$$⁡β $$tan$$⁡γ $$tan$$⁡δ is The value of $$1tan$$⁡α$$+1tan$$⁡β$$+1tan$$⁡γ$$+1tan$$⁡δ is

Question

If α,β,γ,δ are the solutions of the equation $$tan$$⁡θ$$+$$π$$4=$$ $$3$$ $$tan3$$θ, no two of which have equal tangents.The value of $$tan$$⁡α$$+$$$$tan$$⁡β$$+$$$$tan$$⁡γ$$+$$$$tan$$⁡δ isThe value of $$tan$$⁡α $$tan$$⁡β $$tan$$⁡γ $$tan$$⁡δ is The value of $$1tan$$⁡α$$+1tan$$⁡β$$+1tan$$⁡γ$$+1tan$$⁡δ is

Infinity Learn · Accepted Answer

We have $$tan$$⁡θ$$+$$π$$4=3tan$$⁡$$3$$θor $$1+$$$$tan$$⁡θ$$1$$−$$tan$$⁡θ$$=3$$×$$3tan$$⁡θ−$$tan3$$⁡θ$$1$$−$$3tan2$$⁡θ⇒$$1+t1$$−$$t=33t$$−$$t31$$−$$3t2$$ $$(putting$$ $$t=$$$$tan$$⁡θ $$)or$$ $$3t4-6t2+8t-$$ $$1=$$ $$0Hence$$,$$S1$$ : sum of roots $$=t1+t2+t3+t4=0S2$$ : sum of product of roots taken two at a time $$=$$ $$-$$ $$2S3$$ : sum of product of roots taken three at time $$=$$ $$-$$ $$8$$ $$/3S4$$ : product of all roots $$=$$ $$-$$ $$1/31t1+1t2+1t3+1t4=$$∑$$t1t2t3t1t2t3t4=$$−$$8/3$$−$$1/3=8$$

If α,β,γ,δ are the solutions of the equation tan⁡θ+π4= 3 tan3θ, no two of which have equal tangents.The value of tan⁡α+tan⁡β+tan⁡γ+tan⁡δ isThe value of tan⁡α tan⁡β tan⁡γ tan⁡δ is The value of 1tan⁡α+1tan⁡β+1tan⁡γ+1tan⁡δ is

Detailed Solution

Get Expert Academic Guidance – Connect with a Counselor Today!