Question

If $5\tan \mathrm{\theta }=3\mathrm{then}\left(\frac{5\sin \mathrm{\theta }-\cos \mathrm{\theta }}{5\sin \mathrm{\theta }+\cos \mathrm{\theta }}\right)=?$
(a) $\frac{2}{3}$

(b) $\frac{1}{3}$

(c) $\frac{1}{2}$

(d) $\frac{3}{5}$

Answer 1

$$\begin{gathered} 5\tan \theta =3 \\ \Rightarrow 5\times \frac{\sin \theta }{\cos \theta }=3 \\ \Rightarrow 5\sin \theta =3\cos \theta .....\left(1\right) \end{gathered}$$
$$\begin{gathered} \therefore \frac{5\sin \theta -\cos \theta }{5\sin \theta +\cos \theta } \\ =\frac{3\cos \theta -\cos \theta }{3\cos \theta +\cos \theta }\left[\mathrm{Using}\left(1\right)\right] \\ =\frac{2\cos \theta }{4\cos \theta } \\ =\frac{1}{2} \end{gathered}$$

Hence, the correct answer is option (c).