NSEJS #5

$a\cos\theta + b\sin\theta = 3$

$\Rightarrow a^2\cos^2\theta + b^2\sin^2\theta + 2ab\sin\theta\cos\theta = 9 \ldots (1)$

$a\sin\theta - b\cos\theta = 4$

$\Rightarrow a^2\sin^2\theta + b^2\cos^2\theta - 2ab\sin\theta\cos\theta = 16 \ldots (2)$

$(1) + (2) = a^2(\cos^2\theta + \sin^2\theta) + b^2(\cos^2\theta + \sin^2\theta) = a^2 + b^2 = 25$

When $\cos$ and $\sin$ appears interchanged, SQUARE AND ADD.

$\Huge \boxed{\Rightarrow a^2+b^2 = 25}$