If sin4 α + 4cos4 β + 2 = 4 sina cosb; α, β [0, π], then cos(α + β) –cos(α

Engineering

Mathematics

Maximum and Minimum Value of Trigonometric Expression

compound angle Formulae

Question

If sin⁴ α + 4cos⁴ β + 2 = 4 $\sqrt{2}$ sina cosb; α, β $\in$ [0, π], then cos(α + β) –cos(α – β) is equal to

– 1

$- \sqrt{2}$

$\sqrt{2}$

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A.M. $\geq$ G.M.

$\frac{\sin^{4} α + 4 \cos^{4} β + 1 + 1}{4} \geq {(\sin^{4} α . 4 \cos^{4} β . 1 .1)}^{\frac{1}{4}}$

sin⁴ α + 4cos² β + 2 $\geq$ 4 $\sqrt{2}$ sinα cosβ given that sin⁴ α + 4cos⁴ β + 2 = $4 \sqrt{2}$ sinα cosβ

⇒ A.M. = G.M. ⇒ sin⁴ α = 1= 4cos⁴ β

sinα = ±1,cosβ = ± $\frac{1}{\sqrt{2}}$ ,As α,β $\in$ [0, π]

⇒ sinα = 1, cos β = ± $\frac{1}{\sqrt{2}}$

⇒ sinβ = $\frac{1}{\sqrt{2}}$ as β $\in$ [0, π]

cos(α + β)–cos(α–β)=–2 sinα sinβ

$= - \sqrt{2}$