First slide

Properties of ITF

Question

If $\cos^{- 1} x - \cos^{- 1} \frac{y}{2} = α$ then $4 x^{2} - 4 x y \cos α + y^{2}$ is equal to

Moderate

A

$- 4 \sin^{2} α$
B

$4 \sin^{2} α$
C

4
D

$2 \sin 2 α$

Solution

We have,
$\begin{array}{l} \cos^{- 1} x - \cos^{- 1} \frac{y}{2} = α \\ \Rightarrow \cos^{- 1} \{\frac{x y}{2} + \sqrt{1 - x^{2}} \sqrt{1 - \frac{y^{2}}{4}}\} = α \\ \Rightarrow \frac{x y}{2} + \sqrt{1 - x^{2}} \sqrt{1 - \frac{y^{2}}{4}} = \cos α \\ \Rightarrow {(\cos α - \frac{x y}{2})}^{2} = (1 - x^{2}) (1 - \frac{y^{2}}{4}) \\ \Rightarrow \cos^{2} α + \frac{x^{2} y^{2}}{4} - x y \cos α = 1 - x^{2} - \frac{y^{2}}{4} + \frac{x^{2} y^{2}}{4} \\ \Rightarrow x^{2} + \frac{y^{2}}{4} - x y \cos α = \sin^{2} α \\ \Rightarrow 4 x^{2} - 4 x y \cos α + y^{2} = 4 \sin^{2} α \end{array}$

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