# If $a\sin\theta+5\cos\theta=5$ then the value of $5\sin\theta-3\cos\theta$ is equal to

$(a)\;5\qquad(b)\;3\qquad(c)\;4\qquad(d)\;None\;of\;these$

$3\sin\theta=5(1-\cos\theta)$
$\qquad\;\;\;=5\times 2\sin^2\large\frac{\theta}{2}$
$\qquad\;\;\;=\tan\large\frac{\theta}{2}$
$\qquad\;\;\;=\large\frac{3}{5}$
$5\sin\theta-3\cos\theta=5\times\large\frac{2\tan\large\frac{\theta}{2}}{1+\tan^2\large\frac{\theta}{2}}-\frac{3(1-\tan^2\large\frac{\theta}{2})}{1+\tan^2\Large\frac{\theta}{2}}$
$\qquad\qquad\qquad=5\times \large\frac{2\times \large\frac{3}{5}}{1+\Large\frac{9}{25}}-\large\frac{3\times (1-\large\frac{9}{25})}{1+\large\frac{9}{25}}$
$\qquad\qquad\qquad=3$
Hence (b) is the correct option.