Question

If 3cos+5sin=5 then find 5cos-3sin=?

Answer 1

$3 \cos( \alpha ) + 5 \sin( \alpha ) = 5 \\ 9 \cos {}^{2} ( \alpha ) + 25 \sin {}^{2} ( \alpha ) + 30 \cos( \alpha ) \sin( \alpha ) = 25$
Now use
$\sin{}^{2}( \alpha ) + \cos {}^{2} \alpha = 1$
To get
$3 \cos( \alpha ) + 5 \sin( \alpha ) = 5 \\ 9 \cos {}^{2} ( \alpha ) + 25 \sin {}^{2} ( \alpha ) + 30 \cos( \alpha ) \sin( \alpha ) = 25 \\ 9 - 9 \sin{}^{2} \alpha + 25 - 25 \cos{}^{2} \alpha + 30 \cos( \alpha ) \sin( \alpha ) = 25 \\ 9 = 25 \cos{}^{2} \alpha + 9 \sin {}^{2} \alpha - 30 \sin( \alpha ) \cos( \alpha ) \\ 9 = {(5 \cos( \alpha ) - 3 \sin( \alpha ) )}^{2} \\ 5 \cos( \alpha ) - 3 \sin( \alpha ) = + - 3$