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Let
\[
f(x)=\left|\begin{array}{ccc}
\operatorname{cosec} x & \sin x & \operatorname{cosec} e^{2} x+\tan x \sec x \\
\sin ^{2} x & \sin ^{2} x & \sec ^{2} x \\
1 & \sin ^{2} x & \sin ^{2} x
\end{array}\right|
\]
then \( \int_{0}^{\pi / 2} f(x) d x \) equals
(a) \( -\left(\frac{\pi}{4}+\frac{8}{15}\right) \)
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{4}+\frac{1}{5} \)
(d) \( \pi \)
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