If ^-1x-^-1y2=, then 4x^2-4xy+y^2 is equal to

Question

If $\cos^{-1}x-\cos^{-1}\dfrac{y}{2}=\alpha$, then $4x^2-4xy\cos\alpha+y^2$ is equal to

Accepted Answer

$$\cos^{-1}\left(\frac{xy}{2}+\sqrt{(1-x^2)\left(1-\frac{y^2}{4}\right)}\right)=\alpha\ \Rightarrow\ xy+\sqrt{(4-y^2)(1-x^2)}=2\cos\alpha.$$

Squaring: $4-y^2-4x^2+x^2y^2=4\cos^2\alpha-4xy\cos\alpha+x^2y^2$, which simplifies to

$$4x^2+y^2-4xy\cos\alpha=4-4\cos^2\alpha=4\sin^2\alpha.$$

Solutions · 1