If tan 4θ · tan 6θ = 1, where 6θ is an acute angle, then find the value of Cot 5θ.

Question

Accepted Answer

Given: tan 4θ · tan 6θ = 1 and 6θ is an acute angle.

Step 1: Use the identity tan A · tan B = 1 implies A + B = 90° (or π/2 radians) because tan A = cot B.

So, 4θ + 6θ = 90°
=> 10θ = 90°
=> θ = 9°

Step 2: Find Cot 5θ = Cot (5 × 9°) = Cot 45°

We know Cot 45° = 1.

Step 3: Check options, 1 is option 4.

However, the options given are:
1. √3
2. -√3
3. -1
4. 1

So, the correct answer is 1 (option 4).

Double-check:
- tan 4θ = tan 36°
- tan 6θ = tan 54°

tan 36° ≈ 0.7265

tan 54° ≈ 1.3764

Product ≈ 0.7265 × 1.3764 ≈ 1.0 (approximately)

Therefore, the solution is consistent.

Final answer: Cot 5θ = 1.

Answer

Answer

Answer