= sin^2 40° + sin^2 50° + sin^2 (230°+180°) + sin^2 (320°-360°= sin^2 40° + sin^2 50° + sin^2 50° + sin^2 320= sin^2 40° + 2(sin^2 50°) + sin^2 320Using the trigonometric identity sin^2 θ + cos^2 θ = 1, we have= (1 - cos^2 40°) + 2(1 - cos^2 50°) + (1 - cos^2 320°= 3 - cos^2 40° + 2 - 2cos^2 50° + 3 - cos^2 320= 8 - cos^2 40° - 2cos^2 50° - cos^2 320°
Now, we can use the fact that sin(x) = cos(90° - x) to simplify further= 8 - sin^2 50° - 2sin^2 40° - sin^2 40= 8 - 3sin^2 40° - sin^2 50°
Therefore, sin^2 40° + sin^2 50° + sin^2 230° + sin^2 320° = 8 - 3sin^2 40° - sin^2 50°.
= sin^2 40° + sin^2 50° + sin^2 (230°+180°) + sin^2 (320°-360°
= sin^2 40° + sin^2 50° + sin^2 50° + sin^2 320
= sin^2 40° + 2(sin^2 50°) + sin^2 320
Using the trigonometric identity sin^2 θ + cos^2 θ = 1, we have
= (1 - cos^2 40°) + 2(1 - cos^2 50°) + (1 - cos^2 320°
= 3 - cos^2 40° + 2 - 2cos^2 50° + 3 - cos^2 320
= 8 - cos^2 40° - 2cos^2 50° - cos^2 320°
Now, we can use the fact that sin(x) = cos(90° - x) to simplify further
= 8 - sin^2 50° - 2sin^2 40° - sin^2 40
= 8 - 3sin^2 40° - sin^2 50°
Therefore, sin^2 40° + sin^2 50° + sin^2 230° + sin^2 320° = 8 - 3sin^2 40° - sin^2 50°.