If `tan x + cot x=4, "then" : tan^(4)x+cot^(4)x=......`A)110 B)191 C)80 D)194

Options

A. 110
B. 191
C. 80
D. 194

Answer

To solve the problem, we start with the given equation:

\[ \tan x + \cot x = 4 \]

We need to find the value of:

\[ \tan^4 x + \cot^4 x \]

### Step 1: Square the given equation

First, we square both sides of the equation:

\[ (\tan x + \cot x)^2 = 4^2 \]

This expands to:

\[ \tan^2 x + 2 \tan x \cot x + \cot^2 x = 16 \]

### Step 2: Simplify using the identity

We know that \(\tan x \cot x = 1\), so we can substitute this into our equation:

\[ \tan^2 x + 2(1) + \cot^2 x = 16 \]

This simplifies to:

\[ \tan^2 x + \cot^2 x + 2 = 16 \]

### Step 3: Isolate \(\tan^2 x + \cot^2 x\)

Now, we isolate \(\tan^2 x + \cot^2 x\):

\[ \tan^2 x + \cot^2 x = 16 - 2 = 14 \]

### Step 4: Square again

Next, we square \(\tan^2 x + \cot^2 x\):

\[ (\tan^2 x + \cot^2 x)^2 = 14^2 \]

This expands to:

\[ \tan^4 x + 2 \tan^2 x \cot^2 x + \cot^4 x = 196 \]

### Step 5: Substitute \(\tan^2 x \cot^2 x\)

Again, since \(\tan^2 x \cot^2 x = 1\):

\[ \tan^4 x + 2(1) + \cot^4 x = 196 \]

This simplifies to:

\[ \tan^4 x + \cot^4 x + 2 = 196 \]

### Step 6: Isolate \(\tan^4 x + \cot^4 x\)

Now, we isolate \(\tan^4 x + \cot^4 x\):

\[ \tan^4 x + \cot^4 x = 196 - 2 = 194 \]

Thus, the final answer is:

\[ \tan^4 x + \cot^4 x = 194 \]

### Conclusion

The answer is:

\[ \text{D) } 194 \]