We are given the equation
\[
\sec(5^\circ - 2\theta) = \csc(5\theta - 5^\circ).
\]
Now, use the identity for secant and cosecant:
\[
\sec x = \frac{1}{\cos x} \quad \text{and} \quad \csc x = \frac{1}{\sin x}.
\]
Substitute these into the equation:
\[
\frac{1}{\cos(5^\circ - 2\theta)} = \frac{1}{\sin(5\theta - 5^\circ)}.
\]
This implies:
\[
\cos(5^\circ - 2\theta) = \sin(5\theta - 5^\circ).
\]
We know the identity \( \cos x = \sin(90^\circ - x) \). Thus, we can write:
\[
5^\circ - 2\theta = 90^\circ - (5\theta - 5^\circ).
\]
Simplifying this equation:
\[
5^\circ - 2\theta = 90^\circ - 5\theta + 5^\circ.
\]
\[
5^\circ - 2\theta = 95^\circ - 5\theta.
\]
\[
3\theta = 90^\circ.
\]
\[
\theta = 30^\circ.
\]
Now, we need to calculate the value of \( \frac{\sin 3\theta + \tan 2\theta}{\sec 2\theta} \) for \( \theta = 30^\circ \).
Substitute \( \theta = 30^\circ \):
\[
\sin(3 \times 30^\circ) = \sin 90^\circ = 1,
\]
\[
\tan(2 \times 30^\circ) = \tan 60^\circ = \sqrt{3},
\]
\[
\sec(2 \times 30^\circ) = \sec 60^\circ = \frac{2}{\sqrt{3}}.
\]
Now, substitute these values into the given expression:
\[
\frac{\sin 3\theta + \tan 2\theta}{\sec 2\theta} = \frac{1 + \sqrt{3}}{\frac{2}{\sqrt{3}}} = \frac{(1 + \sqrt{3}) \cdot \sqrt{3}}{2}.
\]
Simplify the numerator:
\[
= \frac{\sqrt{3} + 3}{2}.
\]
Thus, the value is \( \frac{1 + \sqrt{3}}{2} \).