What is the exact value of Tan(pi/12)

We have

0 < π/12 < π/2

and

tan(π/12) = sin(π/12) / cos(π/12)

Both sin(x) and cos(x) are positive for 0 < x < π/2, so we expect tan(π/12) to also be positive.

Recall the double-angle identities for cosine,

cos²(x) = (1 + cos(2x)) / 2

as well as the Pythagorean identity,

tan²(x) = sec²(x) - 1

and by definition,

sec(x) = 1 / cos(x)

Putting everything together, we have

tan(x) = √(1 / cos²(x) - 1)

tan(x) = √(2 / (1 + cos(2x)) - 1)

Let x = π/12. Then 2x = π/6, and cos(π/6) = √3 / 2, so that

tan(π/12) = √(2 / (1 + cos(π/6)) - 1)

tan(π/12) = √(2 / (1 + √3 / 2) - 1)

tan(π/12) = √(4 / (2 + √3) - 1)

tan(π/12) = √(4 / (2 + √3) - (2 + √3) / (2 + √3))

tan(π/12) = √((2 - √3) / (2 + √3))

tan(π/12) = √(7 - 4 √3)

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