Answer:
x ∈ {0, π, 2π}
Step-by-step explanation:
The expression can be simplified, which facilitates an answer.
tan(x -π) -cos(x -3π/2) = 0
The tangent function has a period of π, so tan(x-π) = tan(x). The cosine function shifted right by 3π/2 is equivalent to the opposite of the sine function.
tan(x) +sin(x) = 0 . . . . substitute simpler equivalents
sin(x)/cos(x) +sin(x) = 0 . . . . use equivalent for tan(x)
sin(x)(1/cos(x) +1) = 0 . . . . factor out sin(x)
sin(x)(cos(x)+1)/cos(x) . . . . . combine terms to a single fraction
This has solutions where sin(x) = 0 and where cos(x) = -1. Those solutions are integer multiples of π.
x ∈ {0, π, 2π}
