Question:
If $5 \tan \theta=3$ then $\left(\frac{5 \sin \theta-\cos \theta}{5 \sin \theta+\cos \theta}\right)=?$
(a) $\frac{2}{3}$
(b) $\frac{1}{3}$
(c) $\frac{1}{2}$
(d) $\frac{3}{5}$
Solution:
$5 \tan \theta=3$
$\Rightarrow 5 \times \frac{\sin \theta}{\cos \theta}=3$
$\Rightarrow 5 \sin \theta=3 \cos \theta \quad \ldots \ldots(1)$
$\therefore \frac{5 \sin \theta-\cos \theta}{5 \sin \theta+\cos \theta}$
$=\frac{3 \cos \theta-\cos \theta}{3 \cos \theta+\cos \theta} \quad[$ Using $(1)]$
$=\frac{2 \cos \theta}{4 \cos \theta}$
$=\frac{1}{2}$
Hence, the correct answer is option (c).
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