How do you use the sum and difference identity to evaluate #tan105#?

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Answer 1 · 2016-09-08T03:41:27

#tan105˚ = -2 - sqrt(3)#

#tan(105˚)= tan(60˚ + 45˚)#

Use the sum formula #tan(A + B) = (tanA + tanB)/(1 - tanAtanB)#

#tan105˚ = (tan60˚ + tan45˚)/(1- tan60˚tan45˚)#

#tan105˚ = (sqrt(3) + 1)/(1 - sqrt(3)(1))#

#tan105˚ = (sqrt(3) + 1)/(1 - sqrt(3))#

Rationalize the denominator:

#tan105˚ = (sqrt(3) + 1)/(1 - sqrt(3)) xx (1 + sqrt(3))/(1 + sqrt(3))#

#tan105˚ = (sqrt(3) + 1 + sqrt(3) + 3)/(1 - 3)#

#tan105˚ = (4 + 2sqrt(3))/(-2)#

#tan105˚ = -2 - sqrt(3)#

Hopefully this helps!