what is the value of tan 67.5 degree in term of surds in simplest form?

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lfryerda eNotes educator| Certified Educator

To evaluate this trig ratio, we need to look at the angle 67.5 degrees and put that in terms of other angles.  In this case, we see that `67.5=135/2` , which suggests that we use the multiple angle formula for tangent.

`tan2 theta={2tan theta}/{1-tan^2 theta}`    where `theta=67.5`  and `tan 2theta=tan135=-1`

`-1={2 tan theta}/{1-tan^2 theta}`    cross-multiply

`tan^2 theta-1=2 tan theta`   move to one side

`tan^2 theta-2tan theta-1=0`   use quadratic formula

`tan theta={2+-sqrt{4+4}}/2`

`={2+-2sqrt2}/2`

`=1+-sqrt2`   we pick the positive version since in the first quadrant

`=1+sqrt2`

This means that `tan67.5=1+sqrt2`

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