Question

If
$cosec\theta -cot\theta =\frac{1}{2},0<\theta <{90}^{\circ }$,
then $cos\theta$
is equal to

• A. $\frac{5}{3}$
• B. $\frac{3}{5}$
• C. $-\frac{3}{5}$
• D. $-\frac{5}{3}$

Answer 1

The correct option is B

$\frac{3}{5}$

Given:

$cosec\theta -cot\theta =\frac{1}{2}.....\left(1\right)$

Using identity

$cose{c}^2\theta -cot{}^2\theta =1$
$(cosec\theta +\cot \theta )(cosec\theta -cot\theta )=1$

Using (1) we can write,

$(cosec\theta +\cot \theta )\left(\frac{1}{2}\right)=1$

$(cosec\theta +\cot \theta )=2$ ------Eq (2)

On adding (1) and (2),
$2cosec\theta =\frac{1}{2}+2=\frac{5}{2}\Rightarrow cosec\theta =\frac{5}{4}$

$\Rightarrow sin\theta =\frac{4}{5}\Rightarrow si{n}^2\theta =\frac{16}{25}$

$\Rightarrow 1-co{s}^2\theta =\frac{16}{25}\Rightarrow co{s}^2\theta =1-\frac{16}{25}=\frac{9}{25}$
$\Rightarrow cos\theta =\pm \frac{3}{5}$

$\because 0<\theta <{90}^{\circ }$
$\therefore cos\theta =\frac{3}{5}$