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:

Step-by-step solution

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 &#8