Question

If $\text{cosec}\theta =\frac{5}{4}$, then $\sec \theta =......$ and $\cot \theta =.....$

• A. $\frac{5}{3},\frac{3}{4}$
• B. $\frac{3}{5},\frac{4}{3}$
• C. $\frac{7}{5},\frac{1}{4}$
• D. $\frac{8}{7},\frac{1}{5}$

Answer 1

The correct option is A $\frac{5}{3},\frac{3}{4}$

$cosec\theta =\frac{5}{4}=\frac{1}{\sin \theta }\therefore \sin \theta =\frac{4}{5}$
Now, ${\sin }^2\theta +{\cos }^2=1$
${\left(\frac{4}{5}\right)}^2+{\cos }^2\theta =1$
${\cos }^2\theta =1-\frac{16}{25}=\frac{9}{25}$
So $\cos \theta =\frac{3}{5}$
and $\sec \theta =\frac{1}{\cos \theta }=\frac{5}{3}$
Also, $\cot \theta =\frac{\cos \theta }{\sin \theta }$
$=\frac{\frac{3}{5}}{\frac{4}{5}}=\frac{3\times 5}{4\times 5}$
$\therefore \cot \theta =\frac{3}{4}$