Evaluate the expression: 2 (tan 54° / cot 36°) - (cot 41° / tan 49°).

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Step 1: Simplify the expression: 2 (tan 54° / cot 36°) - (cot 41° / tan 49°).

Step 2: Use the identity cot θ = 1 / tan θ.

So, tan 54° / cot 36° = tan 54° / (1 / tan 36°) = tan 54° * tan 36°.

Similarly, cot 41° / tan 49° = (1 / tan 41°) / tan 49° = 1 / (tan 41° * tan 49°).

Step 3: Note that 54° and 36° are complementary angles (54° + 36° = 90°), so tan 54° = cot 36° = 1 / tan 36°.

But let's calculate tan 54° * tan 36°:

Using approximate values:
- tan 54° ≈ 1.37638
- tan 36° ≈ 0.72654

So, tan 54° * tan 36° ≈ 1.37638 * 0.72654 ≈ 1.0

Step 4: For the second term, note that 41° and 49° are complementary angles (41° + 49° = 90°), so tan 49° = cot 41° = 1 / tan 41°.

Therefore, tan 41° * tan 49° = tan 41° * (1 / tan 41°) = 1.

Hence, cot 41° / tan 49° = 1 / (tan 41° * tan 49°) = 1 / 1 = 1.

Step 5: Substitute back:

Expression = 2 * 1 - 1 = 2 - 1 = 1.

Step 6: Re-examine the first term calculation:

Actually, tan 54° * tan 36° = 1 (as shown), so 2 * 1 = 2.

Expression = 2 - 1 = 1.

Step 7: However, the initial approximate calculation suggests the answer is 1.

But let's verify the exact values:

Since tan 54° = cot 36° = 1 / tan 36°, tan 54° * tan 36° = 1.

Similarly, tan 41° * tan 49° = 1.

Therefore, expression = 2 * 1 - 1 = 1.

Final answer: 1.

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