Question

${\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}cos\theta cose{c}^2\theta d\theta =$

• A. $\sqrt{2}-1$
• B. $1-\sqrt{2}$
• C. $\sqrt{2}+1$
• D. $Noneofthese$

Answer 1

The correct option is A $\sqrt{2}-1$

$Let{\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}cos\theta \frac{1}{si{n}^2\theta }d\theta$
$Putt=sin\theta \Rightarrow dt=cos\theta d\theta ,thenwehave$
${\int }_{\frac{1}{\sqrt{2}}}^1\frac{1}{t^2}dt={\left[\frac{-1}{t}\right]}_{\frac{1}{\sqrt{2}}}^1=\sqrt{2}-1$