prove that maximum value of cos⁡αcos⁡β occurs when α=β=σ2If α,β∈(0,π2) and α+β=σ(constant), then prove that maximum value of cos⁡αcos⁡β occurs when α=β=σ2

Hint:As&nbsp;α+β=σ&nbsp;(constant),&nbsp;cos⁡(α+β)=cos⁡(σ)=&nbsp;(constant)So, we need to maximize&nbsp;cos(α-β)

cos⁡αcos⁡β=cos⁡(α+β)+cos⁡(α−β)2=cos⁡σ+cos⁡(σ−2β)2Hence, it is maximized when β=σ2 so that cos⁡(σ−2β)=1

Hint:As&nbsp;α+β=σ&nbsp;(constant),&nbsp;cos⁡(α+β)=cos⁡(σ)=&nbsp;(constant)So, we need to maximize&nbsp;cos(α-β)

cos⁡αcos⁡β=cos⁡(α+β)+cos⁡(α−β)2=cos⁡σ+cos⁡(σ−2β)2Hence, it is maximized when β=σ2 so that cos⁡(σ−2β)=1

"prove that maximum value of cos⁡αcos⁡β occurs when..." was answered by students like you

Averie Ferguson

2022-03-31

prove that maximum value of

\cos α \cos β

occurs when

α = β = \frac{σ}{2}

α, β \in (0, \frac{π}{2})

and

α + β = σ

(constant), then prove that maximum value of

\cos α \cos β

occurs when

α = β = \frac{σ}{2}

Janessa Foster

Beginner2022-04-01Added 12 answers

Hint:
As $α + β = σ$ (constant), $\cos (α + β) = \cos (σ) =$ (constant)
So, we need to maximize $\cos (α - β)$

German Ferguson

Beginner2022-04-02Added 18 answers

\cos α \cos β = \frac{\cos (α + β) + \cos (α - β)}{2} = \frac{\cos σ + \cos (σ - 2 β)}{2}

Hence, it is maximized when

β = \frac{σ}{2}

so that

\cos (σ - 2 β) = 1

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