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

Question

If cos-1 x - cos-1 (y/2) = α, then 4x2 - 4xy cosα + y2 is equal to (A) 4 (B) 2sin2α (C) -4sin2α (D) 4sin2α

Tardigrade Admin · Accepted Answer

Using cos-1A - cos-1 B = cos-1 (AB + √(1 - A2) √( 1 - B2) ∴ cos-1 x - cos-1 (y/2) = α ⇒ (xy/2) + √1 - x2 √ 1 - (y2/4) = cos α ⇒ (cos α - (xy/2))2 = (1 - x2)(1 - (y2/4)) ⇒ 4x2 - 4xy cos α + y2 = 4 (1 - cos2α) = 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

$4$

$2 s in 2 α$

$- 4 s i n^{2} α$

$4 s i n^{2} α$

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

4

2sin2α

−4sin2α

4sin2α

Using cos−1A−cos−1B =cos−1(AB+(1−A2)​(1−B2)​ ∴cos−1x−cos−12y​=α ⇒2xy​+1−x2​1−4y2​​=cosα ⇒(cosα−2xy​)2=(1−x2)(1−4y2​) ⇒4x2−4xycosα+y2=4(1−cos2α)=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

$4$

$2 s in 2 α$

$- 4 s i n^{2} α$

$4 s i n^{2} α$