中四數學問題trigonometry(續三角學)

1.
tan²θ/[1 + tan²(180° - θ)] + sin²(90° + θ)
= tan²θ/[1 + (-tanθ)²] + cos²θ
= tan²θ/[1 + tan²θ] + cos²θ
= (sinθ/cosθ)²/[1 + (sinθ/cosθ)²] + cos²θ
= (sin²θ/cos²θ)/[(cos²θ + sin²θ)/cos²θ] + cos²θ
= (sin²θ/cos²θ)/(1/cos²θ) + cos²θ
= sin²θ + cos²θ
= 1


2.
tanθ + tan(90° - θ) = 3
tanθ + (1/tanθ) = 3
[tanθ + (1/tanθ)]² = 3²
tan²θ + 2(tanθ)(1/tanθ) + (1/tanθ)² = 9
tan²θ + 2 + (1/tan²θ) = 9
Hence, tan²θ + (1/tan²θ) = 7