We are asked to evaluate the following double integral:

$I = \int_{0}^{1} \int_{\sin^{-1}(y)}^{\pi/2} \sec^2{(\cos{x})} \, dx \, dy$

Step 1: Interpret the integral and reverse the order of integration

The limits suggest that $x$ ranges from $\sin^{-1}(y)$ to $\frac{\pi}{2}$, and $y$ ranges from 0 to 1. To simplify the problem, we may reverse the order of integration.

Let's first find the region of integration in the $xy$-plane. The outer integral suggests $y$ runs from 0 to 1. For each $y$, the inner integral suggests that $x$ ranges from $\sin^{-1}(y)$ to $\frac{\pi}{2}$.

This suggests that for the reversed order of integration:

• $x$ will range from 0 to $\frac{\pi}{2}$.
• For each fixed $x$, $y$ will range from 0 to $\sin{x}$ (since $\sin^{-1}(y) \leq x$ implies $y \leq \sin{x}$).

Thus, the integral can be rewritten as:

$I = \int_0^{\frac{\pi}{2}} \int_0^{\sin{x}} \sec^2(\cos{x}) \, dy \, dx$

Step 2: Evaluate the inner integral

The inner integral is with respect to $y$, and the integrand is independent of $y$. Therefore, we can easily evaluate it:

$\int_0^{\sin{x}} \sec^2(\cos{x}) \, dy = \sec^2(\cos{x}) \cdot \sin{x}$

Thus, the double integral becomes:

$I = \int_0^{\frac{\pi}{2}} \sec^2(\cos{x}) \sin{x} \, dx$

Step 3: Simplify the remaining integral

Next, we look at the remaining integral:

$I = \int_0^{\frac{\pi}{2}} \sec^2(\cos{x}) \sin{x} \, dx$

Let us use the substitution $u = \cos{x}$, so that $du = -\sin{x} \, dx$. When $x = 0$, $u = 1$, and when $x = \frac{\pi}{2}$, $u = 0$. The integral becomes:

$I = \int_1^0 \sec^2(u) (-du) = \int_0^1 \sec^2(u) \, du$

Step 4: Evaluate the final integral

The integral of $\sec^2(u)$ is $\tan(u)$, so we have:

$I = \left[ \tan(u) \right]_0^1 = \tan(1) - \tan(0) = \tan(1)$

Final Answer:

The value of the integral is:

$\boxed{\tan(1)}$

Would you like any further clarification or additional steps? Here are some related questions for further exploration:

1. What would happen if the integrand was $\sec^2(\cos{x}) \cos{x}$ instead?
2. How would this change if the limits for $x$ and $y$ were reversed again?
3. How can you use trigonometric identities to simplify integrals like this in general?
4. What substitution could simplify integrals involving $\sec^2(\cos{x})$?
5. How can you apply similar steps to a double integral where the limits of integration are more complicated?

Tip: When dealing with double integrals, always consider whether reversing the order of integration simplifies the process!