The angles A, B, C of a triangle ABC satisfy `4cosAcosB + sin2A + sin2B + sin2C = 4`, Then which of the following statements is/are correct?

Options

A. The triangle ABC is right angled
B. The triangle ABC is isosceles
C. The triangle ABC is neither isosceles nor right angled
D. The triangle ABC is equilateral

Answer

To solve the problem, we need to analyze the given equation involving the angles of triangle ABC:

Given:
\[ 4 \cos A \cos B + \sin 2A + \sin 2B + \sin 2C = 4 \]

### Step 1: Rewrite the equation

We know that:
\[ \sin 2A = 2 \sin A \cos A \]
\[ \sin 2B = 2 \sin B \cos B \]
\[ \sin 2C = 2 \sin C \cos C \]

Substituting these into the equation gives:
\[ 4 \cos A \cos B + 2 \sin A \cos A + 2 \sin B \cos B + 2 \sin C \cos C = 4 \]

### Step 2: Simplify the equation

We can factor out 2 from the last three terms:
\[ 4 \cos A \cos B + 2 (\sin A \cos A + \sin B \cos B + \sin C \cos C) = 4 \]

### Step 3: Analyze the triangle properties

Since A, B, and C are angles of a triangle, we know that:
\[ A + B + C = \pi \]
Thus, we can express C as:
\[ C = \pi - A - B \]

### Step 4: Substitute C into the equation

Using the identity for sine:
\[ \sin C = \sin(\pi - A - B) = \sin(A + B) \]
This leads us to:
\[ \sin C = \sin A \cos B + \cos A \sin B \]

Now substituting this back into our equation:
\[ 4 \cos A \cos B + 2 \left( \sin A \cos A + \sin B \cos B + (\sin A \cos B + \cos A \sin B) \right) = 4 \]

### Step 5: Further simplify

This can be simplified to:
\[ 4 \cos A \cos B + 2 \sin A \cos A + 2 \sin B \cos B + 2 \sin A \cos B + 2 \cos A \sin B = 4 \]

### Step 6: Analyze the maximum values

Since we know that \( \sin C \) must be less than or equal to 1, we can analyze the maximum values of the trigonometric functions involved. The maximum value of \( \cos A \cos B + \sin A \sin B \) occurs when \( A = B \).

### Step 7: Conclude the type of triangle

If \( A = B \), then \( C = \pi - 2A \). For \( C \) to be \( \frac{\pi}{2} \), we must have:
\[ A = B = \frac{\pi}{4} \]
Thus, triangle ABC is a right-angled triangle with angles \( 45^\circ, 45^\circ, 90^\circ \).

### Final Conclusion

From the analysis, we can conclude that:
- The triangle ABC is isosceles (since \( A = B \)).
- The triangle ABC is also right-angled (since \( C = 90^\circ \)).

Thus, the correct statements are:
1. The triangle ABC is a right-angled triangle.
2. The triangle ABC is isosceles.