Statement -1: for any complex number z, `|Re(z)|+|Im(z)| le |z|`
Statement-2: `|sintheta| le 1`, for all `theta`

Options

A. Statement-1 is True, Statement-2 is True: Statement-2 is a correct exp,anation for statement-1.
B. Statement-1 is true, statement -2 is true, Statement-2 is not a correct explanation for statement-1.
C. Statement-1 is True, statement-2 is false,
D. statement-1 is False, Statement-2 is true.

Answer

To solve the problem, we need to analyze the two statements provided:

### Statement 1:
For any complex number \( z \), \( |Re(z)| + |Im(z)| \leq |z| \).

Let’s denote the complex number \( z \) as:
\[ z = x + iy \]
where \( x = Re(z) \) (the real part) and \( y = Im(z) \) (the imaginary part).

The modulus of \( z \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]

Now, we need to evaluate \( |Re(z)| + |Im(z)| \):
\[ |Re(z)| + |Im(z)| = |x| + |y| \]

To prove or disprove the statement, we will use the triangle inequality which states that for any two real numbers \( a \) and \( b \):
\[ |a + b| \leq |a| + |b| \]

In our case, we can apply this to the values \( x \) and \( y \):
\[ |x + iy| = |z| = \sqrt{x^2 + y^2} \]

Using the properties of the triangle inequality, we can say:
\[ |x| + |y| \geq \sqrt{x^2 + y^2} \]

However, the statement claims:
\[ |x| + |y| \leq \sqrt{x^2 + y^2} \]

This is incorrect. Therefore, Statement 1 is **false**.

### Statement 2:
\[ |sin(\theta)| \leq 1 \]

This statement is a well-known property of the sine function. The sine of any angle \( \theta \) is always between -1 and 1, inclusive. Thus, we can write:
\[ -1 \leq sin(\theta) \leq 1 \]

Taking the absolute value gives us:
\[ |sin(\theta)| \leq 1 \]

This statement is **true**.

### Conclusion:
- Statement 1 is false.
- Statement 2 is true.

Thus, the correct option is that Statement 2 is true and Statement 1 is false.

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