How do you use the sum or difference identities to find the exact value of $tan((23pi)/12)$ ?

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Answer 1 · 2016-11-30T12:43:25

$tan((23pi)/12) = frac (1 - sqrt(3)) (1+ sqrt(3))$

First note that $(23pi)/12 =2pi-pi/12$

and as $tan (x+2pi) = tan(x)$ , then:

$tan((23pi)/12) = tan(-pi/12)$

Now: $1/12=1/3-1/4$ ,

So:

$tan(-pi/12) = tan(pi/4-pi/3)$

Using:

$tan(alpha+beta) = frac (tan alpha - tan beta) (1+tan alpha tan beta)$

$tan((23pi)/12) = frac (tan (pi/4) - tan (pi/3)) (1+ tan (pi/4) tan (pi/3))$

$tan((23pi)/12) = frac (1 - sqrt(3)) (1+ sqrt(3))$

How do you use the sum or difference identities to find the exact value of tan((23pi)/12)tan((23pi)/12)?