Introduction:
In a right-angled triangle, one acute angle is double the other. We need to prove that the hypotenuse is double the smallest side. Let's assume the acute angles of the right-angled triangle as 'x' and '2x'.

Proof:
We can start by using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Step 1: Identify the sides of the triangle:
Let's label the sides of the triangle as follows:
- Smallest side: 'a'
- Other side: 'b'
- Hypotenuse: 'c'

Step 2: Apply the Pythagorean theorem:
According to the theorem, we have:
a² + b² = c²

Step 3: Relate the angles to the sides:
We are given that one acute angle is double the other. Let's assume the smaller angle is 'x' and the larger angle is '2x'. Therefore, we can express the sides in terms of these angles as follows:
- Smallest side: a = sin(x)
- Other side: b = sin(2x)
- Hypotenuse: c = cos(2x)

Step 4: Prove that the hypotenuse is double the smallest side:
We need to prove that c = 2a.

Step 4.1: Express a and b in terms of x:
Using the trigonometric identities, we can rewrite a and b as follows:
a = sin(x)
b = sin(2x) = 2sin(x)cos(x)

Step 4.2: Substitute a and b in the Pythagorean theorem:
Substituting the values of a and b in the equation a² + b² = c², we get:
sin²(x) + (2sin(x)cos(x))² = cos²(2x)

Step 4.3: Simplify the equation:
Expanding and simplifying the equation, we get:
sin²(x) + 4sin²(x)cos²(x) = 1 - sin²(2x)

Step 4.4: Apply trigonometric identities:
Using the trigonometric identity sin²(x) + cos²(x) = 1, we can rewrite the equation as:
5sin²(x)cos²(x) = 1 - sin²(2x)

Step 4.5: Apply double angle formula:
Using the double angle formula sin(2x) = 2sin(x)cos(x), we can substitute sin²(2x) in the equation:
5sin²(x)cos²(x) = 1 - (2sin(x)cos(x))²

Step 4.6: Simplify and factor the equation:
Simplifying further, we get:
5sin²(x)cos²(x) = 1 - 4sin²(x)cos²(x)
9sin²(x)cos²(x) = 1

Step 4.7: Apply trigonometric identity:
Using the trigonometric identity