Question

If $3\sin \theta +5\cos \theta =5$, prove that $5\sin \theta -3\cos \theta =\pm 3$.

Answer 1

Given $3\sin \theta +5\cos \theta =5$

Let $5\sin \theta -3\cos \theta =x$

Squaring on both sides for both the equations

$\Rightarrow 9{\sin }^2\theta +25{\cos }^2\theta +30\sin \theta \cos \theta =25$

$\Rightarrow 25{\sin }^2\theta +9{\cos }^2\theta -30\sin \theta \cos \theta =x^2$

Adding the equations ;

$\Rightarrow 34({\sin }^2\theta +{\cos }^2\theta )=25+x^2$

$\Rightarrow x^2=34-25=9$

$\Rightarrow x=\pm 3$

$\therefore 5\sin \theta -3\cos \theta =\pm 3$

Hence proved