Given that tan 45=1, use tan(x+y) to show that tan 22.5= /2 - 1.

(/is square root sign

tan(x+y) = (tanx + tany)/(1 - tanxtany)

so tan 45
= tan(22.5 + tan22.5)/(1 - tan22.5tan22.5)
= 2tan22.5)/1 - tan^2 22.5)
let tan22.5 = y for ease of typing

1 = 2y/(1 - y^2)
1 - y^2 = 2y
y^2 + 2y - 1 = 0
y^2 + 2y = 1 , by completing the square
y^2 + 2y + 1 = 1+1
(y+1)^2 = 2
y+1 = √2
y = √2-1

tan22.5° = √2 - 1

tan(x+y)=(tan(x)+tan(y))/(1-tan(x)tan(y))

substitute x=y to get
tan(2x) = 2tan(x)/(1-tan²(x))
Substitute tan(2x)=1 and solve for tan(x).
Reject the negative root knowing that tan(x)≥0 for 0<x<π/2.

To find the value of tan 22.5° using the tan(x+y) formula, we start by letting x = y = 45°.

Using tan(x+y) = (tan x + tan y) / (1 - tan x * tan y), we have:

tan 45° = (tan 45° + tan 45°) / (1 - tan 45° * tan 45°)

1 = (2tan 45°) / (1 - tan² 45°)

Since tan 45° = 1 (given), we can substitute it into the equation:

1 = (2 * 1) / (1 - 1²)

1 = 2 / (1 - 1)

1 = 2 / 0

This is undefined since division by zero is not possible.

However, using the half-angle formula, we can still find the value of tan 22.5°.

The half-angle formula for tangents states:

tan(2θ) = (2tanθ) / (1 - tan²θ)

Letting θ = 45°, we have:

tan(90°) = (2tan 45°) / (1 - tan² 45°)

Since tan(90°) is undefined, we can manipulate the equation to find the value of tan 22.5°:

(2tan 45°) / (1 - tan² 45°) = tan(90°)

Cross multiplying, we get:

2tan 45° = tan(90°) * (1 - tan² 45°)

Dividing both sides by (1 - tan² 45°), we have:

2tan 45° / (1 - tan² 45°) = tan(90°)

But we know that tan 90° is undefined, so the value of the left side of the equation must also be undefined.

Therefore, using the formula tan(x+y) and the given value of tan 45°, we cannot directly prove that tan 22.5° is √2 - 1.