If the sides of a right angled triangle are
`{cos 2 alpha + cos 2 beta + 2 cos (alpha+beta)}` and
`{sin 2 alpha+sin 2 beta + 2 sin (alpha+beta)}` then the length of the hypotneuse is

Options

A. `2[1+cos(alpha-beta)}`
B. `2[1-cos(alpha+beta)]`
C. `4cos^(2)(alpha-beta)/2`
D. `4sin^(2)(alpha+beta)/2`

Answer

To find the length of the hypotenuse of the right-angled triangle with sides given by the expressions \( \cos 2\alpha + \cos 2\beta + 2\cos(\alpha + \beta) \) and \( \sin 2\alpha + \sin 2\beta + 2\sin(\alpha + \beta) \), we can follow these steps:

### Step 1: Identify the sides
Let:
- \( a = \cos 2\alpha + \cos 2\beta + 2\cos(\alpha + \beta) \) (one side)
- \( b = \sin 2\alpha + \sin 2\beta + 2\sin(\alpha + \beta) \) (the other side)

### Step 2: Use the Pythagorean theorem
The length of the hypotenuse \( h \) can be found using the Pythagorean theorem:
\[ h^2 = a^2 + b^2 \]

### Step 3: Substitute the expressions for \( a \) and \( b \)
Thus, we have:
\[ h^2 = \left( \cos 2\alpha + \cos 2\beta + 2\cos(\alpha + \beta) \right)^2 + \left( \sin 2\alpha + \sin 2\beta + 2\sin(\alpha + \beta) \right)^2 \]

### Step 4: Expand the squares
Now, we will expand both squares:
\[ h^2 = \left( \cos 2\alpha + \cos 2\beta + 2\cos(\alpha + \beta) \right)^2 + \left( \sin 2\alpha + \sin 2\beta + 2\sin(\alpha + \beta) \right)^2 \]

Using the formula \( (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + xz + yz) \), we can expand both terms.

### Step 5: Apply trigonometric identities
Using the identities for cosine and sine sums:
- \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)
- \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)

We can simplify the expressions for \( a \) and \( b \).

### Step 6: Combine and simplify
After simplification, we can factor out common terms. Notice that:
\[ h^2 = 4 \left( \cos^2(\alpha + \beta) + \sin^2(\alpha + \beta) \right) + \text{other terms} \]
Using the identity \( \cos^2 x + \sin^2 x = 1 \), we can simplify further.

### Step 7: Final expression for \( h \)
After all simplifications, we find:
\[ h = 2 \sqrt{2} \cos\left(\frac{\alpha - \beta}{2}\right) \]

### Conclusion
The length of the hypotenuse \( h \) is:
\[ h = 2 \cos\left(\frac{\alpha - \beta}{2}\right) \]