Half angle identity, find Tan (22 1/2)

Question

Using half angle identity and the unit circle to find the value of Tan (22 1/2)

Answer 1

Half angle identity Tanθ=2tan(θ/2)/1-tan^2(θ/2)

If θ=45 then θ/2=45/2=22 1/2

Substitute θ=45

Tan45=2tan(45/2)/1-tan^2(45/2)

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

Let tan(45/2)=x then

1=2x/1-x^2

Multiply each side by 1-x^2

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

1-x^2=2x

x^2+2x-1=0

Add 1 to each side

x^2+2x+1-1=0+1

(x+1)^2-1=1

Add 1 to each side

(x+1)^2-1+1=1+1

(x+1)^2=2

Square root on both sides

Sqrt(x+1)^2=Sqrt2

x+1=Sqrt2

Subtract 1 from each side

x+1-1=Sqrt2-1

x=Sqrt2-1

Substitute x= Tan(45/2)=Tan(22 1/2)

Tan(22 1/2)=Sqrt2-1