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Trigonometric Ratios for Sum of Two Angles

If tanθ+tan...

Question

If $t a n θ + t a n 4 θ + t a n 7 θ = t a n θ t a n 4 θ t a n 7 θ$ , then $θ =$

\frac{n π}{4}

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\frac{n π}{7}

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\frac{n π}{12}

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n π

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Solution

The correct option is B $\frac{n π}{12}$
$tan θ + tan 4 θ + tan 7 θ = tan θ tan 4 θ tan 7 θ$

$tan θ + tan 4 θ = - tan 7 θ [1 - tan θ tan 4 θ]$
$\frac{tan θ + tan 4 θ}{1 - tan θ tan 4 θ} = tan (- 7 θ)$
$\Rightarrow tan 5 θ = tan (- 7 θ)$
$\Rightarrow 5 θ = n π - 7 θ$
$\Rightarrow 12 θ = n π$
$\Rightarrow θ = \frac{n π}{12}, n \in I$

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If $t a n θ + t a n 4 θ + t a n 7 θ = t a n θ t a n 4 θ t a n 7 θ$ , then $θ =$