How do you find the exact values of tan pi/ 8 using the half-angle formula?

We have to find the value of tan (π/8)

Solution

tan π= 180

tan(π/8) = 180/8

tan(π/8) = 22.5

We know the formula that

Tan 2α = 2Tanα /(1 – Tan²α)

Here, α = 22.5 °

Therefore 2α = 2 * 22.5 = 45°

also we have learnt tha Tan 45° = 1

Let us assume Tan 22.5° = x

Substituting in teh above formula α as x we get

=> 1 = 2x/(1 – x²)

=> 1 – x² = 2x

=> x² + 2x – 1 = 0

=> x = (- 2 ± √2² – 4(1)(-1) )/2

=> x = (– 2 ± √8 )/2

=> x = (– 2 ± 2√2 )/2

=> x = – 1 ± √2

=> x = √2 – 1 or -1 – √2

as 22.5° lies in 1st Quadrant and is +ve

so -1 – √2 will be ignored

Hence x = √2 – 1

Answer

Tan 22.5° = √2 – 1