Answer:
We know that:
sin^2a + cos^2a = 1 (1)
Also, we have:
cos a + sin a = 1/2
Squaring both sides of the above equation, we get:
cos^2a + 2cos a sin a + sin^2a = 1/4
Using equation (1), we can simplify this to:
1 + 2cos a sin a = 1/4
Subtracting 1 from both sides, we get:
2cos a sin a = -3/4
Squaring both sides, we get:
4cos^2a sin^2a = 9/16
Using the identity:
sin^2a = 1 - cos^2a
We can rewrite the above equation as:
4cos^2a (1 - cos^2a) = 9/16
Expanding and rearranging, we get:
4cos^4a - 4cos^2a + 9/16 = 0
Multiplying both sides by 16, we get:
64cos^4a - 64cos^2a + 9 = 0
Letting x = cos^2a, we can rewrite this as a quadratic equation:
64x^2 - 64x + 9 = 0
Solving for x using the quadratic formula, we get:
x = (64 ± √16384)/128
x = 1/2 or x = 9/64
Since 0 < a < 2π, we know that cos a is positive, so we can take the positive square root:
cos a = √(1/2) = 1/√2
Substituting this into the equation sin a + cos a = 1/2, we get:
sin a = 1/2 - cos a = 1/2 - 1/√2 = (√2 - 1)/2
Therefore:
sin^2a - cos^2a = ((√2 - 1)/2)^2 - (1/√2)^2
= (3 - 2√2)/4
Therefore, sin^2a - cos^2a = (3 - 2√2)/4.