Question

If $a=\displaystyle \int _{0}^{1}\frac{c o s \left(sin x\right)}{sec ⁡ x}dx,$ then the value of $a^{2}+\left(c o s\right)^{2}\left(sin 1\right)$ is equal to

• A. $0$
• B. $1$
• C. $sin\left(1\right)$
• D. $sin\left(sin 1\right)$

Answer 1

Answer: (B)

$a =\int_{0}^{1} \cos (\sin x) \cos x d x$
Let $\sin x= t$
$\Rightarrow \cos x d x = dt$
$\therefore a =\int_{0}^{\sin 1} \cos t dt =(\sin t)_{0}^{\sin 1}$
$=\sin (\sin 1)-\sin 0$
$=\sin (\sin 1)$
$\therefore a ^{2}+\cos ^{2}(\sin 1)=\sin ^{2} \theta+\cos ^{2} \theta=1 ;(\theta=\sin 1)$