If Sin 4 α + 4 cos4 β + 2 = 4√2 sin α cosβ ; α , β ∈ [0, π] , then cos (α + β) - cos(α - β)is equal to:

Question

If Sin 4 α + 4 cos4 β + 2 = 4√2 sin α cosβ ; α , β ∈ [0, π] , then cos (α + β) - cos(α - β)is equal to: (A) 0 (B) -√2 (C) -1 (D) √2

Tardigrade Admin · Accepted Answer

A.M. ge G.M. ( sin4 α + 4 cos4 β + 1 + 1/4) ge ( sin4α.4 cos4 β.1.1)(1/4) sin4 + 4 cos2β + 2 ge 4√2 sin α cos β given that sin4 α + 4 cos4 β + 2 = 4√2 sinα cosβ ⇒ A.M. = G.M. ⇒ sin4 α = 1 =4 cos4 β sin α = 1, cosβ = ±; (1/√2) ⇒ sin β = (1/√2) as β ∈ [0, π] cos (α + β) - cos(α - β) = -2 sin α sin β = -√2

Q. If Sin $^{4} α + 4 co s^{4} β + 2 = 42 s in α cos β;$
$α, β \in [0, π],$ then cos $(α + β) - cos (α - β)$ is equal to:

$0$

$- 2$

$- 1$

$2$

Q. If Sin 4α+4cos4β+2=42​sinαcosβ; α,β∈[0,π],then cos (α+β)−cos(α−β)is equal to:

0

−2​

−1

2​

A.M.≥G.M. 4sin4α+4cos4β+1+1​≥(sin4α.4cos4β.1.1)41​ sin4+4cos2β+2≥42​sinαcosβ given that sin4α+4cos4β+2=42​sinαcosβ ⇒A.M.=G.M.⇒sin4α=1=4cos4β sinα=1,cosβ=±;2​1​ ⇒sinβ=2​1​asβ∈[0,π] cos(α+β)−cos(α−β)=−2sinαsinβ =−2​

Q. If Sin $^{4} α + 4 co s^{4} β + 2 = 42 s in α cos β;$
$α, β \in [0, π],$ then cos $(α + β) - cos (α - β)$ is equal to:

$0$

$- 2$

$- 1$

$2$