If 3 sin θ + 5 cos θ = 5, then the value of 5 sin θ - 3 cos θ will be
3 sin θ + 5 cos θ = 5
5 sin θ - 3 cos θ = x
On squaring them and adding, we get
$$9 sin^2 \theta + 25 cos^2 \theta + 25 sin^2 \theta + 9 cos^2 \theta = 25 + x^2$$
$$34 = 25 + x^2$$
$$x = 3, -3$$
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