If cos α + cos β = 3 2 and sin α + sin β = 1 2 and θ is the arithmetic mean of α & β , then sin 2 θ + cos 2 θ is equal to:

Question

If cos α + cos β = 3 2 and sin α + sin β = 1 2 and θ is the arithmetic mean of α & β , then sin 2 θ + cos 2 θ is equal to:, 3 5, 7 5, 4 5, 8 5

MARKS App · Accepted Answer

Given cos α + cos β = 3 2 sin α + sin β = 1 2 ⇒ 2 c o s ( α + β 2 ) c o s ( α - β 2 ) = 3 2 ......(i) and 2 sin ⁡ ( α + β 2 ) c o s ⁡ ( α - β 2 )= 1 2 ....(ii) Dividing (ii) and (i) tan α + β 2 ∴ tan ⁡ θ = 1 3 sin 2 θ + cos ⁡ 2 θ = 2 tan ⁡ θ 1 + tan 2 ⁡ θ + 1 - tan 2 ⁡ θ 1 + tan 2 ⁡ θ = 2 1 3 1 + 1 9 + 1 - 1 9 1 + 1 9 = 2 3 + 8 9 10 9 = 14 10 = 7 5

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