Question

Without using trigonometric tables, prove that:

tan 15° tan 60° tan 75° = `sqrt(3)`

Answer 1

Given: tan 15° tan 60° tan 75°

To Prove: tan 15° tan 60° tan 75° = `sqrt(3)`

Proof [Step-wise]:

1. `tan 60^circ = sqrt(3)`.

2. 75° = 90° – 15°

So tan 75° = tan(90° – 15°)

= cot 15°

i.e. tan(90° – θ) = cot θ.

3. Therefore tan 15° · tan 75°

= tan 15° · cot 15° 

= 1

4. Multiply by tan 60°: 

tan 15° · tan 60° · tan 75°

= (tan 15° · tan 75°) · tan 60° 

= `1 · sqrt(3)` 

= `sqrt(3)`

Hence tan 15° · tan 60° · tan 75° = `sqrt(3)`.