First slide

Properties of ITF

Question

If $x = \sin (2 \tan^{- 1} 2)$ , $y = \sin (\frac{1}{2} \tan^{- 1} \frac{4}{3})$ , then

Moderate

A

x = 1 – y
B

x² = 1 – y
C

x² = 1 + y
D

y² = 1 – x

Solution

x = sin2θ, where tanθ = 2
$\Rightarrow x = \frac{2 \tan θ}{1 + \tan^{2} θ} = \frac{4}{1 + 4} = \frac{4}{5}$
If $α = \tan^{- 1} \frac{4}{3}, y = \sin (\frac{α}{2}) = \frac{1}{\sqrt{2}} \sqrt{1 - \cos α}$
$\begin{aligned} = \frac{1}{\sqrt{2}} \sqrt{1 - \frac{3}{5}} = \frac{1}{\sqrt{5}} \\ ∴ y^{2} = \frac{1}{5} \Rightarrow y^{2} = 1 - x \end{aligned}$

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