Question

Tan [x-pi/4] = tanx - 1

Answer 1

Solve for x:
-tan(π/4 - x) = tan(x) - 1
Subtract tan(x) - 1 from both sides:
1 - tan(π/4 - x) - tan(x) = 0
Simplify and substitute y = -tan(x).
1 - tan(π/4 - x) - tan(x) = (-tan(x) (1 - tan(x)))/(-tan(x) - 1)
 = (y (y + 1))/(y - 1):
(y (y + 1))/(y - 1) = 0
Multiply both sides by y - 1:
y (y + 1) = 0
Split into two equations:
y = 0 or y + 1 = 0
Substitute back for y = -tan(x):
-tan(x) = 0 or y + 1 = 0
Multiply both sides by -1:
tan(x) = 0 or y + 1 = 0
Take the inverse tangent of both sides:
x = π n_1 for n_1 element Z
 or y + 1 = 0
Subtract 1 from both sides:
x = π n_1 for n_1 element Z
 or y = -1

Substitute back for y = -tan(x):
x = π n_1 for n_1 element Z
 or -tan(x) = -1
Multiply both sides by -1:
x = π n_1 for n_1 element Z
 or tan(x) = 1
Take the inverse tangent of both sides:
Answer:  x = π n_1 for n_1 element Z  or x = π n_2 + π/4 for n_2 element Z