Question

$\mathrm{If}5\tan \theta =4,\mathrm{then}\mathrm{write}\mathrm{the}\mathrm{value}\mathrm{of}\frac{\left(\cos \theta -\sin \theta \right)}{\left(\cos \theta +\sin \theta \right)}.$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ 5\tan \theta =4 \\ \Rightarrow \tan \theta =\frac{4}{5} \\ \mathrm{Now}, \\ \frac{\left(\cos \theta -\sin \theta \right)}{\left(\cos \theta +\sin \theta \right)} \\ =\frac{\left({\displaystyle \frac{\cos \theta }{\cos \theta }}-{\displaystyle \frac{\sin \theta }{\cos \theta }}\right)}{\left({\displaystyle \frac{\cos \theta }{\cos \theta }}+{\displaystyle \frac{\sin \theta }{\cos \theta }}\right)}\left(\mathrm{Dividing}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\mathrm{by}\cos \theta \right) \\ =\frac{\left(1-\tan \theta \right)}{\left(1+\tan \theta \right)} \\ =\frac{\left({\displaystyle \frac{1}{1}}-{\displaystyle \frac{4}{5}}\right)}{\left({\displaystyle \frac{1}{1}}+{\displaystyle \frac{4}{5}}\right)} \\ =\frac{\left({\displaystyle \frac{1}{5}}\right)}{\left({\displaystyle \frac{9}{5}}\right)} \\ =\frac{1}{9} \end{gathered}$$