Correct Answer - Option 3 : 1/2
Given:
α + β = 90°
α = 2β
Formula used:
Sin(90° - θ) = Cosθ
Tan(90° - θ) = Cotθ
Sec(90° - θ) = Cosecθ
Sin2 A + Cos2 B = 1
then, A + B = 90°
Calculation:
α + β = 90° ----(1)
α = 2β ----(2)
From the equation (1) and (2)
2β + β = 90°
⇒ 3β = 90°
⇒ β = 30°
Now, from (1)
α = 2 × 30°
⇒ α = 60°
Now, we have to find the value of Cos2α + Sin2β
Cos2 60° + Sin2 30°
⇒ 1/4 + 1/4
⇒ 1/2
∴ The value of cos260° + sin230° is 1/2.