If a cos θ + b sin θ = p and a sin θ - b cos θ = q, then the relation between a, b, p and q is

Expression 1 : a cos θ + b sin θ = p

Squaring both sides, we get :

=> $$a^2 cos^2 \theta + b^2 sin^2 \theta + 2ab sin\theta cos\theta = p^2$$ --------Eqn(1)

Expression 2 : a sin θ - b cos θ = q

Squaring both sides, we get :

=> $$a^2 sin^2 \theta + b^2 cos^ \theta - 2ab sin\theta cos\theta = q^2$$ ----------Eqn(2)

Adding eqns (1) & (2)

=> $$a^2 (sin^2 \theta+cos^2 \theta) + b^2 (sin^2 \theta + cos^2 \theta) = p^2 + q^2$$

=> $$a^{2} + b^{2} = p^{2}+ q^{2}$$