Given $3\cos x - 4 \sin x = 2$, find $3 \sin x + 4 \cos x$ without first solving for $x$ If $$3\cos{x}-4\sin{x}=2$$
find $$3\sin{x} +4\cos{x} $$
I have solved the equation for $x$, then calculated the required value, but I think there is a direct solution without solving the equation.

$$3\sin x+4\cos x=A$$
$$3\cos x-4\sin x=2$$
$$9\sin^2x+24\sin x\cos x+16\cos^2x=A^2$$
$$9\cos^2x-24\sin x\cos x+16\sin^2x=4$$
add both equations
$$9+16=A^2+4$$
$$A^2=21$$
$$A=\pm \sqrt{21}$$
