A right-angled triangle with hypotenuse \( h \) has two legs that can be defined using trigonometric functions. Let the angle be \( \theta \) at vertex \( A \) such that \( \angle OAB = \theta \). Then, the lengths of the legs \( OA \) and \( OB \) are given by \( OA = h \cos\theta \) and \( OB = h \sin\theta \).
Step 1: Compute the area
The area \( A \) of the triangle is calculated as follows:
\[
A = \frac{1}{2} \times OA \times OB
\]
\[
= \frac{1}{2} \times h \cos\theta \times h \sin\theta
\]
\[
= \frac{1}{2} h^2 \sin\theta \cos\theta
\]
Utilizing the trigonometric identity \( \sin\theta \cos\theta = \frac{1}{2} \sin 2\theta \), the area becomes:
\[
A = \frac{1}{4} h^2 \sin 2\theta
\]
Step 2: Maximize the area
The maximum value of \( \sin 2\theta \) is 1, which occurs when \( 2\theta = 90^\circ \) (i.e., \( \theta = 45^\circ \)). Consequently, the maximum area \( A_{\text{max}} \) is:
\[
A_{\text{max}} = \frac{1}{4} h^2 \times 1 = \frac{h^2}{4}
\]
Step 3: Verify the correct option
The calculated maximum area is \( \frac{h^2}{4} \), which corresponds to option (D).