Question

Find the value of the following:

tan 15° tan 30° tan 45° tan 60° tan 75°

Answer 1

tan 30° = `1/sqrt(3)`, tan 45° = 1, tan 60° = `sqrt(3)`

tan 15°. tan 30°. tan 45°. tan 60°. tan 75° = `tan 15^circ * 1/sqrt(3) * 1 * sqrt(3) tan 75^circ`

= `tan 15^circ xx tan 75^circ xx 1/sqrt(3) xx 1 xx sqrt(3)`

= `tan(90^circ - 75^circ) xx 1/(cot75^circ) xx 1` ...[tan 90° – θ = cot θ]

= `cot 75^circ xx 1/(cot75^circ) xx 1`

= 1

Answer 2

Step-by-step values:

• tan 15^∘ = 2 − √3​

• tan ⁡30^∘ = `1/sqrt3`

• tan ⁡45^∘ = 1

• tan⁡ 60^∘ = √3

• tan ⁡75^∘ = 2 + √3

`tan 15° xx tan 75° = (2-sqrt3)(2+sqrt3)=4-3=1`

`tan 30°xxtan60° = 1/sqrt3 xx sqrt3 = 1`

tan 45^∘ = 1

1 × 1 × 1

= 1