If cos-1 x-cos-1 (y/2)=α, then 4x2-4xy cos α+y2 is equal to :

Question

If cos-1 x-cos-1 (y/2)=α, then 4x2-4xy cos α+y2 is equal to : (A) -4 sin2 α (B) 4 sin2 α (C) 4 (D) 2 sin 2 α

Tardigrade Admin · Accepted Answer

Key Idea: cos-1 x± cos-1 y=cos-1(xy∓√1-x2√1-y2). We have, cos-1 x-cos-1 (y/2)=α ⇒ cos-1((xy/2)+√1-x2√1-(y2/4))=α ⇒ (xy/2)+√1-x2√1-(y2/4)=cos α ⇒ 2√1-x2√1-(y2/4)=2 cos α-xy ⇒ 4-4x2-y2+x2y2=4 cos2 α+x2y2-2xy cos α ⇒ 4x2-4 xy cos α+y2=4 sin2 α.

Q. If $co s^{- 1} x - co s^{- 1} \frac{y}{2} = α,$ then $4 x^{2} - 4 x y cos α + y^{2}$ is equal to :

$- 4 s i n^{2} α$

$4 s i n^{2} α$

$4$

$2 s in 2 α$

Q. If cos−1x−cos−12y​=α, then 4x2−4xycosα+y2 is equal to :

−4sin2α

4sin2α

4

2sin2α

Key Idea : cos−1x±cos−1y=cos−1(xy∓1−x2​1−y2​). We have, cos−1x−cos−12y​=α ⇒cos−1(2xy​+1−x2​1−4y2​​)=α ⇒2xy​+1−x2​1−4y2​​=cosα ⇒21−x2​1−4y2​​=2cosα−xy ⇒4−4x2−y2+x2y2=4cos2α+x2y2−2xycosα ⇒4x2−4xycosα+y2=4sin2α.

Q. If $co s^{- 1} x - co s^{- 1} \frac{y}{2} = α,$ then $4 x^{2} - 4 x y cos α + y^{2}$ is equal to :

$- 4 s i n^{2} α$

$4 s i n^{2} α$

$4$

$2 s in 2 α$