Giải phương trình: tan(x - pi/4) = - 1, tan(3x + pi/3) = 0, cot(3x - 1) = 1

1. To solve the equation tan(x - π/4) = -1, we can use the inverse tangent function to find the value of x.

Taking the inverse tangent of both sides, we have:

x - π/4 = arctan(-1)

Using the unit circle or a calculator, we find that arctan(-1) = -π/4.

Therefore, x - π/4 = -π/4.

Adding π/4 to both sides, we get:

x = 0.

So the solution to the equation tan(x - π/4) = -1 is x = 0.

2. To solve the equation tan(3x + π/3) = 0, we can use the fact that the tangent function is equal to zero at certain angles.

We know that tan(0) = 0, so we can set 3x + π/3 equal to 0:

3x + π/3 = 0

Subtracting π/3 from both sides, we get:

3x = -π/3

Dividing both sides by 3, we have:

x = -π/9

So the solution to the equation tan(3x + π/3) = 0 is x = -π/9.

3. To solve the equation cot(3x - 1) = 1, we can use the fact that the cotangent function is equal to the reciprocal of the tangent function.

Since cot(θ) = 1/tan(θ), we can rewrite the equation as:

1/tan(3x - 1) = 1

Multiplying both sides by tan(3x - 1), we get:

1 = tan(3x - 1)

Using the inverse tangent function, we have:

3x - 1 = arctan(1)

Using the unit circle or a calculator, we find that arctan(1) = π/4.

Therefore, 3x - 1 = π/4.

Adding 1 to both sides, we get:

3x = π/4 + 1

Simplifying, we have:

3x = (π + 4)/4

Dividing both sides by 3, we have:

x = (π + 4)/12

So the solution to the equation cot(3x - 1) = 1 is x = (π + 4)/12.