First slide

Properties of ITF

Question

$C o s^{- 1} (\frac{x}{a}) + C o s^{- 1} (\frac{y}{b}) = θ$ $t h e n$ $\frac{x^{2}}{a^{2}} - \frac{2 x y}{a b} \cos θ + \frac{y^{2}}{b^{2}} =$

Moderate

A

$\sin^{2} θ$
B

$\cos^{2} θ$
C

$\tan^{2} θ$
D

$c o t^{2} θ$

Solution

$\cos^{- 1} \frac{x}{a} + \cos^{- 1} \frac{y}{b} = θ \Rightarrow \cos^{- 1} (\frac{x}{a} \frac{y}{b} - \sqrt{1 - {(\frac{x}{a})}^{2}} \sqrt{1 - {(\frac{y}{b})}^{2}}) = θ \Rightarrow \frac{x}{a} \frac{y}{b} - \sqrt{1 - {(\frac{x}{a})}^{2}} \sqrt{1 - {(\frac{y}{b})}^{2}} = \cos θ \Rightarrow \frac{x}{a} \frac{y}{b} - \cos θ = \sqrt{1 - {(\frac{x}{a})}^{2}} \sqrt{1 - {(\frac{y}{b})}^{2}} \Rightarrow {(\frac{x}{a} \frac{y}{b} - \cos θ)}^{2} = (1 - {(\frac{x}{a})}^{2}) (1 - {(\frac{y}{b})}^{2}) \Rightarrow \frac{x^{2}}{a^{2}} - 2 \frac{x y}{a b} \cos θ + \frac{y^{2}}{b^{2}} = 1 - \cos^{2} θ = \sin^{2} θ$

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