To prove that tan 75° cot 75° = 4, we start with the definitions of tangent and cotangent. We know that:
- tan θ = sin θ / cos θ
- cot θ = 1 / tan θ = cos θ / sin θ
Thus, we can write:
tan 75° cot 75° = tan 75° × cot 75° = tan 75° × (1 / tan 75°) = 1
This expression simplifies to 1, which seems to contradict our goal of proving it equals 4. However, there is a small error in interpreting the relationship between tangent and cotangent for the specific angles involved.
Instead, let’s use the identities for tangent:
- tan(75°) = tan(45° + 30°)
Using the tangent addition formula:
tan(A + B) = (tan A + tan B) / (1 – tan A tan B
Where A = 45° and B = 30°:
- tan(45°) = 1
- tan(30°) = 1/√3
Inserting these values:
tan(75°) = (1 + 1/√3) / (1 – 1 × (1/√3))
Now, simplify:
tan(75°) = (√3 + 1) / (√3 – 1) = (√3 + 1) (√3 + 1) / ((√3 – 1)(√3 + 1)) = (3 + 2√3 + 1) / (2) = (4 + 2√3) / 2 = 2 + √3
We now have tan(75°) = 2 + √3. We also know that cot(75°) is the reciprocal of tan(75°):
cot(75°) = 1 / tan(75°) = 1 / (2 + √3) = (2 – √3) / 1 = 2 – √3.
Now multiplying tan(75°) and cot(75°):
tan 75° cot 75° = (2 + √3)(2 – √3) = 4 – 3 = 1,
Therefore, tan(75°) cot(75°) equals 1, not 4. So, we conclude that the statement in the question is incorrect as tan 75° cot 75° actually equals 1.