First slide

Properties of ITF

Question

$I f θ = \tan^{- 1} (2 \tan^{2} θ) - \tan^{- 1} (\frac{1}{3} \tan θ)$ , then $\tan θ =$

Moderate

A

$- 2$
B

$- 1$
C

$\frac{2}{3}$
D

$2$

Solution

$W e h a v e θ = \tan^{- 1} \frac{2 \tan^{2} θ - \frac{1}{3} \tan θ}{1 + \frac{2}{3} \tan^{3} θ}$
$\begin{array}{l} \Rightarrow \tan θ = \frac{6 \tan^{2} θ - \tan θ}{3 + 2 \tan^{3} θ} \\ \Rightarrow 1 = \frac{6 \tan θ - 1}{3 + 2 \tan^{3} θ} or \tan θ = 0 \\ \Rightarrow 2 \tan^{3} θ - 6 \tan θ + 4 = 0 \\ \Rightarrow {(\tan θ - 1)}^{2} (\tan θ + 2) = 0 \\ \Rightarrow \tan θ = 1; \tan θ = - 2; \tan θ = 0 \end{array}$

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