If $y=\cos \left(\frac{\pi}{3}+\cos ^{-1} \frac{x}{2}\right)$, then $(x-y)^{2}+3 y^{2}$ is equal to ______.

JEE Main · 2025 · 02 Apr 2025 · Shift 2 · Inverse Trigonometric Functions

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

, then

(x - y)^{2} + 3 y^{2}

is equal to ______.

Answer: 3

With

φ = cos^{- 1} \frac{x}{2}

cos φ = \frac{x}{2}

and

sin φ = \frac{4 - x ^{2}}{2}

. Then

y = cos (\frac{π}{3} + φ) = \frac{1}{2} \cdot \frac{x}{2} - \frac{3}{2} \cdot \frac{4 - x ^{2}}{2}

, so

4 y = x - 3 4 - x^{2}

. Writing

x - 4 y = 3 4 - x^{2}

and squaring gives

x^{2} - 8 x y + 16 y^{2} = 3 (4 - x^{2})

, which simplifies to

x^{2} - 2 x y + 4 y^{2} = 3

. Since

x^{2} - 2 x y + 4 y^{2} = (x - y)^{2} + 3 y^{2}

, the value is

3

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