Question

How do you use the sum and difference identities to find the exact value of tan 105 degrees?

Answer 1

The special triangles, `30:60:90 and 45:45:90`, allow us to evaluate sine and cosine and tangent.

We leverage that information to evaluate tan(105).

`tan(a+b)=(tan(a)+tan(b))/(1-tan(a)tan(b))`

`tan(60+45)=(tan(60)+tan(45))/(1-tan(60)tan(45))`

`tan(105)=(tan(60)+tan(45))/(1-tan(60)tan(45))`

`tan(105)=(sqrt(3)+1)/(1-sqrt(3)*1)`

`tan(105)=(sqrt(3)+1)/(1-sqrt(3))`

Rationalize

`tan(105)=(sqrt(3)+1)/(1-sqrt(3))*(1+sqrt(3))/(1+sqrt(3))`

`tan(105)=(sqrt(3)+3+1+sqrt(3))/(1-3)`

`tan(105)=(2sqrt(3)+4)/(-2)`

`tan(105)=-3.732050808`

To evaluate make sure that the calculator is in Degree mode.