Consider the following statements
I. `tan theta` increases faster than `sintheta " as " theta` increases.
II. The value of `sin theta + cos theta` is always greater than 1.
Which of the statements given above is/are correct ?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

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

**Statement I:** `tan(theta)` increases faster than `sin(theta)` as `theta` increases.

**Statement II:** The value of `sin(theta) + cos(theta)` is always greater than 1.

Let's evaluate each statement step by step.

### Step 1: Evaluate Statement I

1. **Understanding `tan(theta)` and `sin(theta)`**:
- Recall that `tan(theta) = sin(theta) / cos(theta)`. As `theta` increases, `sin(theta)` increases, but `cos(theta)` decreases until it reaches zero at `theta = 90°`, where `tan(theta)` becomes undefined.

2. **Calculating Values**:
- For `theta = 0°`:
- `sin(0°) = 0`, `tan(0°) = 0`
- For `theta = 30°`:
- `sin(30°) = 1/2`, `tan(30°) = 1/√3 ≈ 0.577`
- For `theta = 45°`:
- `sin(45°) = 1/√2 ≈ 0.707`, `tan(45°) = 1`
- For `theta = 60°`:
- `sin(60°) = √3/2 ≈ 0.866`, `tan(60°) = √3 ≈ 1.732`
- For `theta = 90°`:
- `sin(90°) = 1`, `tan(90°)` is undefined.

3. **Analysis**:
- As `theta` increases from `0°` to `90°`, `tan(theta)` starts off equal to `sin(theta)` but then increases without bound while `sin(theta)` approaches 1. Thus, `tan(theta)` indeed increases faster than `sin(theta)`.

**Conclusion for Statement I**: **True**

### Step 2: Evaluate Statement II

1. **Understanding `sin(theta) + cos(theta)`**:
- We need to determine if `sin(theta) + cos(theta) > 1` for all values of `theta`.

2. **Calculating Values**:
- For `theta = 0°`:
- `sin(0°) + cos(0°) = 0 + 1 = 1`
- For `theta = 30°`:
- `sin(30°) + cos(30°) = 1/2 + √3/2 ≈ 1.366`
- For `theta = 45°`:
- `sin(45°) + cos(45°) = 1/√2 + 1/√2 = √2 ≈ 1.414`
- For `theta = 60°`:
- `sin(60°) + cos(60°) = √3/2 + 1/2 = (√3 + 1)/2 ≈ 1.366`
- For `theta = 90°`:
- `sin(90°) + cos(90°) = 1 + 0 = 1`

3. **Analysis**:
- The value of `sin(theta) + cos(theta)` equals 1 at `theta = 0°` and `theta = 90°`, but it does not exceed 1 at these points. Therefore, the statement that it is always greater than 1 is incorrect.

**Conclusion for Statement II**: **False**

### Final Conclusion

- **Statement I** is correct.
- **Statement II** is incorrect.

Thus, the answer is that **only the first statement is correct**.