Use a halfangle identity to find the exact value of the foll

Use a half-angle identity to find the exact value of the following expression tan 22.5 degree Which of the following is the correct value of tan 22 5 degree? 1- Squareroot 2 1+ Squareroot 2 -1-Squareroot 2 -1 + Squareroot 2

Solution

Here we have that

tan 2x= 2tan x/(1- tan^2x)

so if x= 22.5 then

tan 45 = 2tan(22.5)/(1-tan^2(22.5)

or 1= 2tan(22.5)/(1-tan^2(22.5)

or 1-tan^2(22.5) =2tan(22.5)

or tan^2(22.5) +2tan(22.5)-1=0

or using quadratic formula, we get

tan(22.5) = (-b +- root(b^2-4ac))/2a = (-2 +- root((-2)^2-4(1)(-1))/2(1) = -2+-\\sqrt{8)/2

Now on taking + sign only, we get

tan(22.5) = -2+2root(2)/2) = -1 + root(2)

So option D is correct answer.