Question

Without using trigonometric tables, find the value of the expression:

3 cot 31° tan 15° cot 27° tan 75° cot 63° cot 59°

Answer 1

Given: 3 cot 31° tan 15° cot 27° tan 75° cot 63° cot 59°

Step-wise calculation:

1. Compute tan 15° · tan 75°: 

`tan 15^circ = 2 - sqrt(3)`

`tan 75^circ = 2 + sqrt(3)`

So `tan 15^circ · tan 75^circ = (2 - sqrt(3))(2 + sqrt(3))`

= 4 – 3

= 1

2. Use complementary-angle identities:

cot θ = tan(90° – θ)   ...(Equivalently tan(90° – θ) = cot θ)

Hence cot 31° = tan 59°

cot 27° = tan 63° 

cot 63° = tan 27° 

cot 59° = tan 31° 

So cot 31° · cot 27° · cot 63° · cot 59°

= tan 59° · tan 63° · tan 27° · tan 31° 

= (tan 59° · tan 31°) · (tan 63° · tan 27°) 

= 1·1

= 1, because tan α · tan(90° – α) = 1.

3. Multiply all factors:

3 × (cot 31° · cot 27° · cot 63° · cot 59°) × (tan 15° · tan 75°) 

= 3 × 1 × 1

= 3

The value of the expression is 3.