Question

$\varphi$ is an acute angle such that $\tan \varphi =\frac{2}{3}$ then evaluate $\left(\frac{1+\tan \varphi }{\sin \varphi +\cos \varphi }\right).\left(\frac{1-\cot \varphi }{\sec \varphi +\csc \varphi }\right)$

• A. $-\frac{1}{5}$
• B. $-\frac{4}{\sqrt{13}}$
• C. $\frac{1}{5}$
• D. $\frac{4}{\sqrt{13}}$

Answer 1

Answer: (A)

$-\frac{1}{5}$

$=\left(\frac{1+\frac{2}{3}}{\frac{2}{\sqrt{13}}+\frac{3}{\sqrt{13}}}\right)\left(\frac{1-\frac{3}{2}}{\frac{\sqrt{13}}{2}+\frac{\sqrt{13}}{3}}\right)=\left(\frac{\frac{5}{3}}{\frac{5}{\sqrt{13}}}\right)\left(\frac{\frac{-1}{2}}{\frac{5\sqrt{13}}{6}}\right)$
$=\frac{\sqrt{13}}{3}\times \left(\frac{-1}{2}\right)\times \frac{6}{5\sqrt{13}}=-\frac{1}{5}.$