To solve the equation tan(3x - π/5) = cot(x), we can use the identity tan(x) = 1/cot(x).

First, let's rewrite the equation using the identity:

tan(3x - π/5) = 1/tan(x)

Now, we can rewrite tan(3x - π/5) as sin(3x - π/5)/cos(3x - π/5) and tan(x) as sin(x)/cos(x):

sin(3x - π/5)/cos(3x - π/5) = 1/(sin(x)/cos(x))

Next, let's simplify the right side by multiplying both sides by cos(x):

sin(3x - π/5) = cos(x)/sin(x)

Now, let's simplify the left side by using the angle subtraction formula for sine:

sin(3x)cos(π/5) - cos(3x)sin(π/5) = cos(x)/sin(x)

Using the angle addition formula for cosine, we can rewrite cos(3x) and sin(3x):

[cos(x)cos(2x) - sin(x)sin(2x)]cos(π/5) - [sin(x)cos(2x) + cos(x)sin(2x)]sin(π/5) = cos(x)/sin(x)

Expanding and simplifying, we get:

cos(x)cos(2x)cos(π/5) - sin(x)sin(2x)cos(π/5) - sin(x)cos(2x)sin(π/5) - cos(x)sin(2x)sin(π/5) = cos(x)/sin(x)

Now, let's simplify further by using the double angle formulas for cosine and sine:

cos(x)(2cos^2(x) - 1)cos(π/5) - sin(x)(2sin(x)cos(x))cos(π/5) - sin(x)(1 - 2sin^2(x))sin(π/5) - cos(x)(2sin(x)cos(x))sin(π/5) = cos(x)/sin(x)

Expanding and simplifying, we get:

2cos^3(x)cos(π/5) - cos(x)cos(π/5) - 2sin^3(x)sin(π/5) + sin(x)sin(π/5) = cos(x)/sin(x)

Now, let's simplify further by using the angle sum formulas for cosine and sine:

2cos^3(x)cos(π/5) - cos(x)cos(π/5) - 2sin^3(x)sin(π/5) + sin(x)sin(π/5) = cos(x)/sin(x)

2cos^3(x)cos(π/5) - cos(x)cos(π/5) - 2sin^3(x)sin(π/5) + sin(x)sin(π/5) = cos(x)/sin(x)

Now, we can simplify the equation further by dividing both sides by cos(x)/sin(x):

2cos^3(x)cos(π/5) - cos(x)cos(π/5) - 2sin^3(x)sin(π/5) + sin(x)sin(π/5) = 1

Now, we have a trigonometric equation that can be solved using algebraic methods. However, the solution is quite complex and involves multiple steps. It is recommended to use a graphing calculator or numerical methods to find the approximate solutions.