If and , then which one of the following is true?

The problem involves solving a system of trigonometric equations to find a relationship between the constants a, b, and c. Given the equations a cos θ + b sin θ = c and a sin θ - b cos θ = d, we can apply the method of squaring and adding. Squaring the first equation yields a² cos² θ + b² sin² θ + 2ab sin θ cos θ = c². Squaring the second equation yields a² sin² θ + b² cos² θ - 2ab sin θ cos θ = d². Adding these two results eliminates the cross-product term (2ab sin θ cos θ). Factoring out a² and b² gives a²(cos² θ + sin² θ) + b²(sin² θ + cos² θ) = c² + d². Using the fundamental trigonometric identity sin² θ + cos² θ = 1, the expression simplifies to a² + b² = c² + d². Rearranging this identity leads to the conclusion that a² + b² - c² = d².