Q. 355.0( 1 Vote )

# If cos^{–1}α + cos^{–1}β + cos^{–1}γ = 3π, then α(β + γ) + β (γ + α) + γ (α + β) equals

A. 0

B. 1

C. 6

D. 12

Answer :

Given, cos^{–1}α + cos^{–1}β + cos^{–1}γ = 3π … (1)

Principal value of cos^{-1} x is [0, π]

So, maximum value which cos^{-1} x can have is π.

So, if (1) is correct then all the three terms i.e,

cos^{–1}α, cos^{–1}β, cos^{–1}γ should be equal to π

So, cos^{–1}α = π

cos^{–1}β = π

cos^{–1}γ = π

So, α = β = γ = -1

So, α(β + γ) + β (γ + α) + γ (α + β)

= (-1)(-1-1) + (-1)(-1-1) + (-1)(-1-1)

= 3(-1)(-2)

= 6

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