IF tanA+cotA=5, then `tan^4 A+cot^4 A` is equal

Options

A. 520
B. 521
C. 529
D. 527

Answer

To solve the problem where \( \tan A + \cot A = 5 \) and we need to find \( \tan^4 A + \cot^4 A \), we can follow these steps:

### Step 1: Square the given equation
We start with the equation:
\[ \tan A + \cot A = 5 \]
Now, we square both sides:
\[ (\tan A + \cot A)^2 = 5^2 \]
This gives us:
\[ \tan^2 A + 2 \tan A \cot A + \cot^2 A = 25 \]

### Step 2: Use the identity for \( \tan A \cot A \)
We know that:
\[ \tan A \cot A = 1 \]
Thus, substituting this into our equation:
\[ \tan^2 A + 2(1) + \cot^2 A = 25 \]
This simplifies to:
\[ \tan^2 A + \cot^2 A + 2 = 25 \]
Subtracting 2 from both sides gives:
\[ \tan^2 A + \cot^2 A = 23 \]

### Step 3: Square again to find \( \tan^4 A + \cot^4 A \)
Now we square \( \tan^2 A + \cot^2 A \):
\[ (\tan^2 A + \cot^2 A)^2 = 23^2 \]
This expands to:
\[ \tan^4 A + 2 \tan^2 A \cot^2 A + \cot^4 A = 529 \]

### Step 4: Substitute \( \tan^2 A \cot^2 A \)
Since \( \tan A \cot A = 1 \), we have:
\[ \tan^2 A \cot^2 A = 1 \]
Thus, substituting this into our equation:
\[ \tan^4 A + 2(1) + \cot^4 A = 529 \]
This simplifies to:
\[ \tan^4 A + \cot^4 A + 2 = 529 \]
Subtracting 2 from both sides gives:
\[ \tan^4 A + \cot^4 A = 527 \]

### Final Answer
Thus, the value of \( \tan^4 A + \cot^4 A \) is:
\[ \boxed{527} \]