If-1-tan-4-1-tan-4-tan-then-find-possible-value-of-tan-11-

Question Number 144914 by bobhans last updated on 30/Jun/21

If \frac{1 + \tan 4 \sqrt{θ}}{1 - \tan 4 \sqrt{θ}} = \tan θ, then find possible

value of \tan (θ + 11 \sqrt{θ}) .

Answered by imjagoll last updated on 30/Jun/21

Answered by mr W last updated on 30/Jun/21

\frac{\tan \frac{π}{4} + \tan 4 \sqrt{θ}}{1 - \tan \frac{π}{4} \times \tan 4 \sqrt{θ}} = \tan θ

\tan (\frac{π}{4} + 4 \sqrt{θ}) = \tan θ

k π + \frac{π}{4} + 4 \sqrt{θ} = θ

k π + \frac{π}{4} + 4 = {(\sqrt{θ} - 2)}^{2}

\sqrt{θ} = 2 + \sqrt{k π + \frac{π}{4} + 4}

\Rightarrow k ⩾ - 1

\sqrt{θ} = 2 - \sqrt{k π + \frac{π}{4} + 4}

\Rightarrow k = - 1

θ + 11 \sqrt{θ} = k π + \frac{π}{4} + 15 \sqrt{θ}

θ + 11 \sqrt{θ} = k π + \frac{π}{4} + 30 \pm 15 \sqrt{k π + \frac{π}{4} + 4}

\Rightarrow \tan (θ + 11 \sqrt{θ}) = \tan (\frac{π}{4} + 30 + 15 \sqrt{k π + \frac{π}{4} + 4}) w i t h k ⩾ - 1

o r

\Rightarrow \tan (θ + 11 \sqrt{θ}) = \tan (\frac{π}{4} + 30 - 15 \sqrt{- π + \frac{π}{4} + 4})

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