Question

Find the value of tan `pi/8`.

Answer 1

`pi/8 = 180^circ/8 = 45^circ/2 = 22 1/2`

We know that tan 2A = `(2 tan "A")/(1 - tan^2 "A")`

Put A = `22 1/2` in the above formula

We get tan 2 `(22 1^circ/2) = (2 tan 22 1^circ/2)/(1 - tan^2 22 1^circ/2)`

`tan 45^circ = (2 tan 22 1^circ/2)/(1 - tan^2 22 1^circ/2)`

`1 = (2 tan 22 1^circ/2)/(1 - tan^2 22 1^circ/2)`

On cross multiplication we get

`1 - tan^2 22 1^circ/2 = 2 tan 22 1^circ/2`

(or) `tan^2 22 1^circ/2 + 2 tan 22 1^circ/2 - 1 = 0`

x = `(-b +- sqrt(b^2 - 4ac))/(2a)`

`tan 22 1^circ/2 = (- 2 +- sqrt(4 - 4 xx 1 xx (-1)))/(2 xx 1)`

Here a = 1, b = 2, c = -1

`= (- 2 +- sqrt(4 + 4))/2`

`= (- 2 +- 2sqrt 2)/2`

`= 2 [(- 1 +- sqrt2)/2] = - 1 +- sqrt2`

Since `22 1/2` is acute tan `22 1/2` is positive tan `22 1/2 = tan pi/8`

= -1 + `sqrt2`

= `sqrt2` – 1

Answer 2

`pi/8 = 180^circ/8 = 45^circ/2 = 22 1/2`

Consider `tan^2  "A"/2 = (sin^2  "A"/2)/(cos^2  "A"/2) = ((1 - cos"A")/2)/((1 + cos "A")/2)`

`(because sin^2 "A" = (1 - cos 2"A")/2; cos^2"A" = (1 + cos 2"A")/2)`

`tan^2  "A"/2 = (1 - cos 2"A")/(1 + cos 2"A")`

Put A = 45°, we get

`tan^2  45^circ/2 = (1 - cos 45^circ)/(1 + cos 45^circ)`

`= (1 - 1/sqrt2)/(1 + 1/sqrt2) = (((sqrt2 - 1)/sqrt2))/(((sqrt2 + 1)/sqrt2))`

`= (sqrt2 - 1)/(sqrt2 + 1) xx (sqrt2 - 1)/(sqrt2 - 1)`

`= (sqrt2 - 1)^2/((sqrt2)^2 - 1^2)`

`tan^2  45^circ/2 = (sqrt2 - 1)^2/1`

`therefore tan^2 22 1/2 = (sqrt2 - 1)^2`

Taking square root, `tan^2 22 1/2 = +- (sqrt2 - 1)`

But `22 1/2` lies in first quadrant, tan `22 1/2` is positive.

`tan 22 1/2 = sqrt2 - 1`

Answer 3

consider tan A = `(sin 2"A")/(1 + cos 2"A")`

Put A = `22 1/2`

`[because (sin 2"A")/(1 + cos 2"A") = (2 sin "A" cos "A")/(3 cos^2 "A") = (sin "A")/(cos "A") = tan A]`

`tan 22 1^circ/2 = (sin (2 xx 22 1^circ/2))/(1 + cos (2 xx 22 1^circ/2))`

`= (sin 45^circ)/(1 + cos 45^circ) = (1/sqrt2)/(1 + 1/sqrt2) = (1/sqrt2)/((sqrt2 + 1)/sqrt2)`

`= (1/sqrt2) xx ((sqrt2)/(sqrt2 + 1))`

`= 1/(sqrt2 + 1) xx (sqrt2 - 1)/(sqrt2 - 1)`

`= (sqrt2 - 1)/((sqrt2)^2 - (1)^2) = (sqrt2 - 1)/(2 - 1)`

tan `22 1/2 = sqrt2 - 1`