A line makes angles α , β , γ with the coordinate axes. If α + β = π 2 , then cos α + cos β + cos γ 2 is equal to

Question

A line makes angles α , β , γ with the coordinate axes. If α + β = π 2 , then cos α + cos β + cos γ 2 is equal to, 1 + sin 2 α, 1 - sin 2 α, 1 + cos 2 α, None of these

MARKS App · Accepted Answer

We have, cos 2 α + cos 2 β + cos 2 γ = 1 ⇒ cos 2 α + cos 2 π 2 - α + cos 2 γ = 1 [given α + β = π 2 ] ⇒ cos 2 α + sin 2 α + cos 2 γ = 1 ⇒ 1 + cos 2 γ = 1 ⇒ cos 2 γ = 0 ⇒ cos γ = 0 ∴ cos α + cos β + cos γ 2 = cos α + sin α 2 = 1 + 2 sin α · cos α = 1 + sin 2 α

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