Which of the following is/are true?

Options

A. `tan abs(tan^(-1)x)=absx`
B. `cot abs(cot^(-1)x)=x`
C. `tan^(-1)abs(tanx)=absx`
D. `sinabs(sin^(-1)x)=absx`

Answer

To determine which of the given options regarding inverse trigonometric functions are true, we will analyze each option step by step.

### Step-by-Step Solution:

**Option 1:**
We need to check if \( \tan(\lvert \tan^{-1}(x) \rvert) = \tan(\tan^{-1}(\lvert x \rvert)) \).

1. **Understanding \( \tan^{-1}(x) \)**:
- For \( x > 0 \), \( \tan^{-1}(x) \) is positive.
- For \( x < 0 \), \( \tan^{-1}(x) \) is negative.

2. **Taking the modulus**:
- Thus, \( \lvert \tan^{-1}(x) \rvert = \tan^{-1}(x) \) if \( x > 0 \).
- \( \lvert \tan^{-1}(x) \rvert = -\tan^{-1}(x) \) if \( x < 0 \).

3. **Evaluating \( \tan(\lvert \tan^{-1}(x) \rvert) \)**:
- For \( x > 0 \): \( \tan(\lvert \tan^{-1}(x) \rvert) = \tan(\tan^{-1}(x)) = x \).
- For \( x < 0 \): \( \tan(\lvert \tan^{-1}(x) \rvert) = \tan(-\tan^{-1}(x)) = -x \).

4. **Evaluating \( \tan(\tan^{-1}(\lvert x \rvert)) \)**:
- Regardless of the sign of \( x \), \( \tan(\tan^{-1}(\lvert x \rvert)) = \lvert x \rvert \).

5. **Conclusion for Option 1**:
- Both expressions yield \( \lvert x \rvert \), hence Option 1 is **True**.

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**Option 2:**
We need to check if \( \tan^{-1}(\lvert x \rvert) = \lvert \tan^{-1}(x) \rvert \).

1. **Understanding \( \tan^{-1}(x) \)**:
- For \( x \geq 0 \), \( \tan^{-1}(x) \) is non-negative.
- For \( x < 0 \), \( \tan^{-1}(x) \) is negative.

2. **Taking the modulus**:
- Thus, \( \lvert \tan^{-1}(x) \rvert = \tan^{-1}(x) \) if \( x \geq 0 \).
- \( \lvert \tan^{-1}(x) \rvert = -\tan^{-1}(x) \) if \( x < 0 \).

3. **Conclusion for Option 2**:
- Since both sides are equal in both cases, Option 2 is **True**.

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**Option 3:**
We need to check if \( \lvert \tan(x) \rvert = \tan(\lvert x \rvert) \).

1. **Understanding \( \tan(x) \)**:
- The tangent function is periodic and has different signs in different quadrants.

2. **Evaluating \( \lvert \tan(x) \rvert \)**:
- It depends on the angle \( x \) and can be positive or negative.

3. **Evaluating \( \tan(\lvert x \rvert) \)**:
- If \( x \) is positive, \( \tan(\lvert x \rvert) = \tan(x) \).
- If \( x \) is negative, \( \tan(\lvert x \rvert) = \tan(-x) = -\tan(x) \).

4. **Conclusion for Option 3**:
- The equality does not hold for all \( x \), hence Option 3 is **False**.

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**Option 4:**
We need to check if \( \lvert \sin^{-1}(x) \rvert = \sin^{-1}(\lvert x \rvert) \).

1. **Understanding \( \sin^{-1}(x) \)**:
- For \( x \in [0, 1] \), \( \sin^{-1}(x) \) is non-negative.
- For \( x \in [-1, 0] \), \( \sin^{-1}(x) \) is negative.

2. **Taking the modulus**:
- Thus, \( \lvert \sin^{-1}(x) \rvert = \sin^{-1}(x) \) if \( x \geq 0 \).
- \( \lvert \sin^{-1}(x) \rvert = -\sin^{-1}(x) \) if \( x < 0 \).

3. **Evaluating \( \sin^{-1}(\lvert x \rvert) \)**:
- It is always non-negative since \( \lvert x \rvert \) is non-negative.

4. **Conclusion for Option 4**:
- Both sides are equal, hence Option 4 is **True**.

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### Final Answers:
- Options that are true: **1, 2, and 4**.