What is the equation to the straight line passing through the point (–sinθ, cosθ) and perpendicular to the line x cosθ + y sinθ = 9?

The given line has slope –cosθ/sinθ. A perpendicular line has slope sinθ/cosθ. Its equation through (–sinθ, cosθ) is: y – cosθ = (sinθ/cosθ)(x + sinθ), i.e., cosθ·y – cos²θ = sinθ·x + sin²θ, so sinθ·x – cosθ·y + sin²θ + cos²θ = 0, giving x sinθ – y cosθ + 1 = 0.