Which of the following quantities is/are positive ? a.`cos (tan^(-1) (tan 4))` b.`sin(cot^(-1)(cot 4))` c.`tan (cos^(-1) (cos 5))` d.`cot(sin^(-1) (sin 4))`

Options

A. `cos (tan^(-1) (tan 4))`
B. `sin(cot^(-1)(cot 4))`
C. `tan (cos^(-1) (cos 5))`
D. `cot(sin^(-1) (sin 4))`

Explanation

To determine which of the given quantities are positive, we will evaluate each expression step by step.

### Step 1: Evaluate \( \cos(\tan^{-1}(\tan 4)) \)

1. **Understanding the expression**: The expression \( \tan^{-1}(\tan 4) \) gives us an angle whose tangent is \( \tan 4 \). However, since \( 4 \) radians is not within the principal range of the arctangent function (which is \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \)), we need to adjust it.

2. **Adjustment**: We can express \( 4 \) in terms of \( \pi \):
\[ 4 \text{ radians} \approx 1.273 \pi \text{ radians} \quad (\text{since } \pi \approx 3.14) \]
Thus, \( \tan^{-1}(\tan 4) = 4 - \pi \).

3. **Calculate \( \cos(4 - \pi) \)**:
\[ \cos(4 - \pi) = -\cos(4) \]
Since \( \cos(4) \) is positive, \( -\cos(4) \) is negative. Therefore, \( \cos(\tan^{-1}(\tan 4)) < 0 \).

### Step 2: Evaluate \( \sin(\cot^{-1}(\cot 4)) \)

1. **Understanding the expression**: The expression \( \cot^{-1}(\cot 4) \) gives us an angle whose cotangent is \( \cot 4 \). Similar to the previous case, \( 4 \) is outside the principal range of the arccotangent function.

2. **Adjustment**: Thus, we have:
\[ \cot^{-1}(\cot 4) = 4 - \pi \]

3. **Calculate \( \sin(4 - \pi) \)**:
\[ \sin(4 - \pi) = -\sin(4) \]
Since \( \sin(4) \) is positive, \( -\sin(4) \) is negative. Therefore, \( \sin(\cot^{-1}(\cot 4)) < 0 \).

### Step 3: Evaluate \( \tan(\cos^{-1}(\cos 5)) \)

1. **Understanding the expression**: The expression \( \cos^{-1}(\cos 5) \) gives us an angle whose cosine is \( \cos 5 \). Since \( 5 \) radians is also outside the principal range of the arccosine function, we adjust it:
\[ \cos^{-1}(\cos 5) = 2\pi - 5 \]

2. **Calculate \( \tan(2\pi - 5) \)**:
\[ \tan(2\pi - 5) = \tan(-5) = -\tan(5) \]
Since \( \tan(5) \) is positive, \( -\tan(5) \) is negative. Therefore, \( \tan(\cos^{-1}(\cos 5)) < 0 \).

### Step 4: Evaluate \( \cot(\sin^{-1}(\sin 4)) \)

1. **Understanding the expression**: The expression \( \sin^{-1}(\sin 4) \) gives us an angle whose sine is \( \sin 4 \). Again, \( 4 \) is outside the principal range of the arcsine function, so we adjust it:
\[ \sin^{-1}(\sin 4) = \pi - 4 \]

2. **Calculate \( \cot(\pi - 4) \)**:
\[ \cot(\pi - 4) = -\cot(4) \]
Since \( \cot(4) \) is positive, \( -\cot(4) \) is negative. Therefore, \( \cot(\sin^{-1}(\sin 4)) < 0 \).

### Conclusion

After evaluating all four expressions, we find that all of them are negative. Therefore, none of the quantities are positive.