Dealing with Half-Angle Identities Involving Radicals
By adding, subtracting, or doubling angle measures, you can find lots of exact values of trigonometry functions. For example, you can use the half-angle identity when the exact value of the trig function involves radicals.
This example uses tan π/8:
Determine which angle is double the angle you’re working with.
Substitute the angle measure into one of the half-angle tangent identities.
Fill in the function values and simplify the answer.
To get the radical out of the denominator, rationalize it by multiplying both parts of the fraction by the radical.
The other identity for the tangent of a half angle gives you exactly the same answer. That form isn’t any easier, though, because both the sine and cosine of this angle have a radical in them. If the problem involved an angle of 60 degrees, though, the story would be different. The sine of 60 degrees is
and the cosine is 1/2 , which practically begs you to use the form with the cosine in the denominator so you don’t have to mess with a radical in the denominator. Both identities work — the one you use is just a matter of personal preference.