Which of the following is correct for any tow complex numbers `z_1a n dz_2?` `|z_1z_2|=|z_1||z_2|` (b) `a r g(z_1z_2)=a r g(z_1)a r g(z_2)` (c) `|z_1+z_2|=|z_1|+|z_2|` (d) `|z_1+z_2|geq|z_1|+|z_2|`

Options

A. `|z_(1) z_(2)| = | z_(1) ||z_(2)|`
B. arg` (z _(1) z_(2)) = arg (z_(1). Arg (z_(2))`
C. `|z_(1) + z_(2)| = |z_(1)|+|z_(2)|`
D. `|z_(1) + z_(2) ge |z_(1)|- | z_(2)|`

Explanation

To solve the question, we need to analyze each of the given statements about complex numbers \( z_1 \) and \( z_2 \) and determine which one is correct.

### Step-by-Step Solution:

1. **Statement (a):** \( |z_1 z_2| = |z_1| |z_2| \)

This statement is true. The modulus of the product of two complex numbers is equal to the product of their moduli. This property is well-established in complex number theory.

**Hint:** Recall the property of moduli for multiplication of complex numbers.

2. **Statement (b):** \( \arg(z_1 z_2) = \arg(z_1) \arg(z_2) \)

This statement is false. The argument of the product of two complex numbers is equal to the sum of their arguments, not the product. Therefore, \( \arg(z_1 z_2) = \arg(z_1) + \arg(z_2) \).

**Hint:** Remember the rule for the argument of a product of complex numbers.

3. **Statement (c):** \( |z_1 + z_2| = |z_1| + |z_2| \)

This statement is also false. The equality holds only when \( z_1 \) and \( z_2 \) are in the same direction in the complex plane. In general, the triangle inequality states that \( |z_1 + z_2| \leq |z_1| + |z_2| \).

**Hint:** Consider the triangle inequality in the context of complex numbers.

4. **Statement (d):** \( |z_1 + z_2| \geq |z_1| + |z_2| \)

This statement is false. The correct interpretation of the triangle inequality is \( |z_1 + z_2| \leq |z_1| + |z_2| \). The inequality states that the modulus of the sum is less than or equal to the sum of the moduli.

**Hint:** Review the triangle inequality for complex numbers.

### Conclusion:
The only correct statement among the options provided is:

**(a)** \( |z_1 z_2| = |z_1| |z_2| \)