The value of sin² 30 - sin² 40 + sin² 45 - sin² 55 - sin² 35 + sin² 45 - sin²…

To solve the expression sin² 30° - sin² 40° + sin² 45° - sin² 55° - sin² 35° + sin² 45° - sin² 50° + sin² 60°, we can simplify it using trigonometric identities, specifically the complementary angle identity: sin(90° - θ) = cos(θ), which implies sin(90° - θ) = cos(θ), and the fundamental identity sin²(θ) + cos²(θ) = 1.

Step-by-Step Simplification
List the terms:
sin² 30° - sin² 40° + sin² 45° - sin² 55° - sin² 35° + sin² 45° - sin² 50° + sin² 60°

Use complementary angles to convert terms:

sin² 55° = cos² 35°

sin² 50° = cos² 40°

Now the expression becomes:
sin² 30° - sin² 40° + sin² 45° - cos² 35° - sin² 35° + sin² 45° - cos² 40° + sin² 60°

Group and simplify using sin²(θ) + cos²(θ) = 1:

Group 1: -(sin² 40° + cos² 40°) = -1

Group 2: -(cos² 35° + sin² 35°) = -1

Constant terms: sin² 30° + sin² 45° + sin² 45° + sin² 60°

Substitute the values of the constant terms:

sin² 30° = (1/2)² = 1/4

sin² 45° = (1/√2)² = 1/2

sin² 60° = (√3/2)² = 3/4

Total sum = (1/4) + (1/2) + (1/2) + (3/4) - 1 - 1
Total sum = (1/4 + 3/4) + (1/2 + 1/2) - 2
Total sum = 1 + 1 - 2 = 0