Which of the following pairs of functions is NOT identical? (a) `e^((lnx)/2)` and `sqrt(x)` (b) `tan(tanx)` and `cot(cotx)` (c) `cos^(2)x+sin^(4)x` and `sin^(2)x+cos^(4)x` (d) `(|x|)/x` and `sgn(x)` where `sgn(x)` stands for signum function.

Options

A. (a) `e^((lnx)/2)` and `sqrt(x)`
B. (b) `tan(tanx)` and `cot(cotx)`
C. (c) `cos^(2)x+sin^(4)x` and `sin^(2)x+cos^(4)x`
D. (d) `(|x|)/x` and `sgn(x)` where `sgn(x)` stands for signum function.

Explanation

To determine which of the following pairs of functions is NOT identical, we will analyze each option step by step.

### Step 1: Analyze Option (a) `e^(ln(x)/2)` and `sqrt(x)`

1. **Rewrite the first function**:
\[ e^{\frac{\ln(x)}{2}} = e^{\ln(x^{1/2})} = x^{1/2} = \sqrt{x} \]
Therefore, both functions are identical.

### Step 2: Analyze Option (b) `tan(tan(x))` and `cot(cot(x))`

1. **Determine the domain of `tan(tan(x))`**:
- The function `tan(x)` is undefined for \( x = \frac{\pi}{2} + n\pi \) where \( n \) is any integer.
- Therefore, `tan(tan(x))` will also be undefined for values of \( x \) that lead to \( tan(x) = \frac{\pi}{2} + n\pi \).

2. **Determine the domain of `cot(cot(x))`**:
- The function `cot(x)` is undefined for \( x = n\pi \) where \( n \) is any integer.
- Thus, `cot(cot(x))` will also be undefined for values of \( x \) that lead to \( cot(x) = 0 \).

Since the domains of `tan(tan(x))` and `cot(cot(x))` are different, these functions are NOT identical.

### Step 3: Analyze Option (c) `cos^2(x) + sin^4(x)` and `sin^2(x) + cos^4(x)`

1. **Check the values at specific points**:
- At \( x = 0 \):
\[ f_1(0) = \cos^2(0) + \sin^4(0) = 1 + 0 = 1 \]
\[ f_2(0) = \sin^2(0) + \cos^4(0) = 0 + 1 = 1 \]

- At \( x = \frac{\pi}{2} \):
\[ f_1\left(\frac{\pi}{2}\right) = \cos^2\left(\frac{\pi}{2}\right) + \sin^4\left(\frac{\pi}{2}\right) = 0 + 1 = 1 \]
\[ f_2\left(\frac{\pi}{2}\right) = \sin^2\left(\frac{\pi}{2}\right) + \cos^4\left(\frac{\pi}{2}\right) = 1 + 0 = 1 \]

- Since both functions yield the same results for all \( x \), they are identical.

### Step 4: Analyze Option (d) `|x|/x` and `sgn(x)`

1. **Define the functions**:
- The function \( \frac{|x|}{x} \) is defined as:
- \( 1 \) for \( x > 0 \)
- \( -1 \) for \( x < 0 \)
- Undefined for \( x = 0 \)

- The signum function \( sgn(x) \) is defined as:
- \( 1 \) for \( x > 0 \)
- \( -1 \) for \( x < 0 \)
- \( 0 \) for \( x = 0 \)

Since \( \frac{|x|}{x} \) is undefined at \( x = 0 \) while \( sgn(x) \) is defined at \( x = 0 \), these functions are NOT identical.

### Conclusion

The pairs of functions that are NOT identical are:
- Option (b): `tan(tan(x))` and `cot(cot(x))`
- Option (d): `|x|/x` and `sgn(x)`

Thus, the answer to the question is that options (b) and (d) are NOT identical.