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 $$=$$ $$-$$ $$1/31t1+1t2+1t3+1t4=$$∑$$t1t2t3t1t2t3t4=$$−$$8/3$$−$$1/3=8$$

Trigonometric Identities

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If $α, β, γ, δ$ are the solutions of the equation $\tan (θ + \frac{π}{4}) =$ 3 tan3 $θ$ , no two of which have equal tangents.

The value of $\tan α + \tan β + \tan γ + \tan δ$ is

The value of $\tan α \tan β \tan γ \tan δ$ is

The value of $\frac{1}{\tan α} + \frac{1}{\tan β} + \frac{1}{\tan γ} + \frac{1}{\tan δ}$ is

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Trigonometric Identities

Remember concepts with our Masterclasses.

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 is

The value of tan⁡α tan⁡β tan⁡γ tan⁡δ is