Which of the following is/are true ?

Options

A. `tan1 gt tan^(-1)1`
B. `sin1 gt cos1`
C. `tan1 lt sin1`
D. `cos(cos1) gt (1)/(sqrt(2))`

Answer

To determine which of the statements are true, we will evaluate each statement step by step.

### Step 1: Evaluate the first statement
**Statement 1:** \( \tan(1) > \tan^{-1}(1) \)

1. **Calculate \( \tan(1) \)**:
- Since \( 1 \) radian is approximately \( 57.3^\circ \), we find \( \tan(1) \) which is \( \tan(57.3^\circ) \).
- The value of \( \tan(57.3^\circ) \) is greater than \( 1 \).

2. **Calculate \( \tan^{-1}(1) \)**:
- \( \tan^{-1}(1) = 45^\circ = \frac{\pi}{4} \) radians, which is approximately \( 0.785 \).

3. **Comparison**:
- Since \( \tan(1) > 1 \) and \( \tan^{-1}(1) \approx 0.785 \), we conclude that \( \tan(1) > \tan^{-1}(1) \) is true.

### Step 2: Evaluate the second statement
**Statement 2:** \( \sin(1) > \cos(1) \)

1. **Using the identity**:
- This implies \( \tan(1) > 1 \) since \( \tan(1) = \frac{\sin(1)}{\cos(1)} \).

2. **Comparison**:
- From the previous evaluation, we know \( \tan(1) \) is indeed greater than \( 1 \).

3. **Conclusion**:
- Therefore, \( \sin(1) > \cos(1) \) is also true.

### Step 3: Evaluate the third statement
**Statement 3:** \( \tan(1) < \sin(1) \)

1. **Rearranging the inequality**:
- We can express this as \( \frac{\sin(1)}{\cos(1)} < \sin(1) \).

2. **Dividing both sides by \( \sin(1) \)** (assuming \( \sin(1) \neq 0 \)):
- This leads to \( \frac{1}{\cos(1)} < 1 \), which implies \( \cos(1) > 1 \).

3. **Conclusion**:
- Since \( \cos(1) \) can never be greater than \( 1 \) (its maximum value is \( 1 \)), this statement is false.

### Step 4: Evaluate the fourth statement
**Statement 4:** \( \cos(\cos(1)) > \frac{1}{2} \)

1. **Calculate \( \cos(1) \)**:
- The value of \( \cos(1) \) is approximately \( 0.54 \).

2. **Calculate \( \cos(0.54) \)**:
- We find \( \cos(0.54) \) which is approximately \( 0.85 \).

3. **Comparison**:
- Since \( 0.85 > 0.5 \), we conclude that \( \cos(\cos(1)) > \frac{1}{2} \) is true.

### Final Conclusion
- **True Statements**:
- Statement 1: True
- Statement 2: True
- Statement 3: False
- Statement 4: True

### Summary of True Statements
- The true statements are 1, 2, and 4.