If y = cos ( 3 + cos – 1 x 2 ) , then ( x – y ) 2 + 3 y 2 is equal to [2025]

Question

If $y = \cos (\frac{π}{3} + \cos^{- 1} \frac{x}{2})$ , then ${(x - y)}^{2} + 3 y^{2}$ is equal to [2025]

Smart Achievers · Accepted Answer

(3)

We have, $y = \cos (\frac{π}{3} + \cos^{- 1} \frac{x}{2})$

$= \cos \frac{π}{3} \cos (\cos^{- 1} \frac{x}{2}) - \sin \frac{π}{3} \sin (\cos^{- 1} \frac{x}{2})$

$= \frac{1}{2} \times \frac{x}{2} - \frac{\sqrt{3}}{2} \sin (\sin^{- 1} \sqrt{1 - \frac{x^{2}}{4}})$

$= \frac{x}{4} - \frac{\sqrt{3}}{2} \frac{\sqrt{4 - x^{2}}}{2}$

$\Rightarrow 4 y = x - \sqrt{3 (4 - x^{2})} \Rightarrow (x - 4 y) = \sqrt{3 (4 - x^{2})}$

On squaring both sides, we get

$x^{2} + 16 y^{2} - 8 x y = 12 - 3 x^{2}$

$\Rightarrow 4 x^{2} + 16 y^{2} - 8 x y = 12 \Rightarrow x^{2} + 4 y^{2} - 2 x y = 3$

$∴ {(x - y)}^{2} + 3 y^{2} = 3$

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