If sin 2 β is the geometric mean between sin α and cos α , then cos 4 β is equal to

Question

If sin 2 β is the geometric mean between sin α and cos α , then cos 4 β is equal to (A) 2(sin)2 ((π /4) - α ) (B) 2(cos)2 ((π /4) - α ) (C) 2(cos)2 ((π /2) + α ) (D) 2(sin)2 ((π /4) + α )

Tardigrade Admin · Accepted Answer

sin2 2 β =sin â¡ α cos â¡ α cos 4 β =1-2sin2 â¡ 2 β =sin2 α +cos2 â¡ α -2sin â¡ α cos â¡ α =(sin α - cos â¡ α )2 =(√2 (sin α ⋅ (1/√2) - cos α (1/√2)))2 =2(sin α cos â¡ (π /4) - cos â¡ α sin â¡ (π /4))2 =2(sin (α - (π /4)))2=2(sin)2 â¡ ((π /4) - α )

Q. If $s in 2 β$ is the geometric mean between $s in α$ and $cos α,$ then $cos 4 β$ is equal to

$2 (s in)^{2} (\frac{π}{4} - α)$

$2 (cos)^{2} (\frac{π}{4} - α)$

$2 (cos)^{2} (\frac{π}{2} + α)$

$2 (s in)^{2} (\frac{π}{4} + α)$

Q. If sin2β is the geometric mean between sinα and cosα, then cos4β is equal to

2(sin)2(4π​−α)

2(cos)2(4π​−α)

2(cos)2(2π​+α)

2(sin)2(4π​+α)

sin22β=sin⁡αcos⁡α cos4β=1−2sin2⁡2β =sin2α+cos2⁡α−2sin⁡αcos⁡α =(sinα−cos⁡α)2 =(2​(sinα⋅2​1​−cosα2​1​))2 =2(sinαcos⁡4π​−cos⁡αsin⁡4π​)2 =2(sin(α−4π​))2=2(sin)2⁡(4π​−α)

Q. If $s in 2 β$ is the geometric mean between $s in α$ and $cos α,$ then $cos 4 β$ is equal to

$2 (s in)^{2} (\frac{π}{4} - α)$

$2 (cos)^{2} (\frac{π}{4} - α)$

$2 (cos)^{2} (\frac{π}{2} + α)$

$2 (s in)^{2} (\frac{π}{4} + α)$