If θ is not integral multiple of π2 and 2A, 3A are not odd multiple of π2 and prove that i) Tanθ+2Tan2θ+4Tan4θ+8Cot8θ=Cotθii) Tan3A⋅Tan2A⋅TanA=Tan3A−Tan2A−TanA

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If $θ$ is not integral multiple of $\frac{π}{2}$ and $2 A, 3 A$ are not odd multiple of $\frac{π}{2}$ and prove that

i) $Tan θ + 2 Tan 2 θ + 4 Tan 4 θ + 8 Co t 8 θ = Cot θ$

ii) $Tan 3 A \cdot Tan 2 A \cdot TanA = Tan 3 A - Tan 2 A - TanA$

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Detailed Solution

i) We know that $Tan A = Cot A - 2 Cot 2 A$

L.H.S $= Tan θ + 2 Tan 2 θ + 4 Tan 4 θ + 8 \cot 8 θ$

$= \cot θ - 2 \cot 2 θ + 2 \cot 2 θ - 4 \cot 4 θ + 4 \cot 4 θ - 8 \cot 8 θ + 8 \cot 8 θ$

$= \cot θ$ = R.H.S

ii) $Tan 3 A = Tan (2 A + A) = \frac{Tan 2 A + Tan A}{1 - Tan 2 A Tan A}$

$Tan 3 A - Tan 3 A Tan 2 A Tan A = Ta n 2 A + Tan A$

$Tan 3 A - Tan 2 A - Tan A = Tan 3 A Tan 2 A Tan A$

$∴$ $Tan 3 A Tan 2 A Tan A = Tan 3 A - Tan 2 A - Tan A$

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