sin θ = a + b, cos θ = a - b

Multiply the two,

sin θ cos θ = (a + b)(a - b)

sin θ cos θ = a² - b² (you should know this property from sec 2)

2 sin θ cos θ = 2(a² - b²)

sin 2θ = 2(a² - b²) (recall double angle formula for sine)

(Shown)

sin θ = a + b , cos θ = a - b

Add the two,

sin θ + cos θ = a + b + a - b
sin θ + cos θ = 2a

Square both sides,

(sin θ + cos θ)² = (2a)²
sin² θ + 2 sin θ cos θ + cos² θ = 4a²
1 + sin 2θ = 4a² ①


If you do subtraction,

sin θ - cos θ = a + b - (a - b)
sin θ - cos θ = 2b

Square both sides,

(sin θ - cos θ)² = (2b)²
sin² θ - 2 sin θ cos θ - cos² θ = 4b²
1 - sin 2θ = 4b² ②


① + ②


1 + sin 2θ + 1 - sin 2θ = 4a² + 4b²
2 = 4(a² + b²)
a² + b² = 2/4 = ½

Alternative for b)

sin θ = a + b
sin² θ = (a + b)²
sin² θ = a² + 2ab + b²

cos θ = a - b
cos² θ = (a - b)²
cos² θ = a² - 2ab + b²


sin² θ + cos² θ = a² + 2ab + b² + a² - 2ab + b²

1 = 2a² + 2b²
a² + b² = ½

c)

If a/b = 3, then a = 3b


tan θ = sin θ / cos θ

tan θ = (a + b)/(a - b)

tan θ = (3b + b)/(3b - b)

tan θ = 4b/2b = 2