If y=cos (π/3+cos -1 x/2 ) , then (x-y) 2 +3 y 2 is equal to

Mathematics · Inverse Trigonometric Functions

JEE Main 2025 — 2 April, Evening Shift — Question 44

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

Numerical answer — enter this value.

$y=\cos \left(\frac{\pi}{3}+\cos ^{-1} \frac{x}{2}\right)$

\begin{aligned} & =\cos \left(\frac{\pi}{3}\right) \cos \left(\cos ^{-1}\left(\frac{x}{2}\right)\right)-\sin \left(\frac{\pi}{3}\right) \sin \left(\cos ^{-1}\left(\frac{x}{2}\right)\right) \\& =\frac{1}{2} \cdot \frac{x}{2}-\frac{\sqrt{3}}{2} \cdot \sqrt{1-\frac{x^{2}}{4}} \\& \Rightarrow 4 y=x-\sqrt{3} \sqrt{4-x^{2}} \\& \Rightarrow(4 y-x)^{2}=3\left(4-x^{2}\right) \\& \Rightarrow 16 y^{2}+x^{2}-8 x y=12-3 x^{2} \\& x^{2}+4 y^{2}-2 x y=3 \\& (x-y)^{2}+3 y^{2}=3 \end{aligned}

Answer key and solution verified before publishing.

Start with this question, then two more from the same chapter — with a tutor that explains every step. Free.