What is the half-angle of tangent 22.5 (degrees)?

Find tan (22.5)

#-1 + sqrt2#

Explanation:

Call tan (22.5) = tan t --> tan 2t = tan 45 = 1

Use trig identity: # tan 2t = (2tan t)/(1 - tan^2 t)# (1)

#tan 2t = 1 = (2tan t)/(1 - tan^2 t)# -->
--> #tan^2 t + 2(tan t) - 1 = 0#
Solve this quadratic equation for tan t.

#D = d^2 = b^2 - 4ac = 4 + 4 = 8# --> #d = +- 2sqrt2#
There are 2 real roots:
tan t = -b/2a +- d/2a = -2/1 + 2sqrt2/2 = - 1 +- sqrt2

#tan t = tan (22.5) = - 1 +- sqrt2#
Since tan 22.5 is positive, then take the positive
tan (22.5) = - 1 + sqrt2

Since 22.522.5 is not an angle where the values of the six trigonometric functions are known, try using half-angle identities.22.522.5 is not an exact angleFirst, rewrite the angle as the product of 1212 and an angle where the values of the six trigonometric functions are known. In this case, 22.522.5 can be rewritten as (12)⋅4512⋅45.tan((12)⋅45)tan12⋅45Use the half-angle identity for tangent to simplify the expression. The formula states that tan(θ2)=sin(θ)1+cos(θ)tanθ2=sinθ1+cosθ.sin(45)1+cos(45)sin451+cos45Simplify the result.
√2−1

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