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

1 Answer
Sep 8, 2016

#tan105˚ = -2 - sqrt(3)#

Explanation:

#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!