Answer:
x = 2 π n_1 for n_1 element Z
or x = π n_2 for n_2 element Z
Explanation:
Solve for x:
tan(x) = sin(x)
Subtract sin(x) from both sides:
tan(x) - sin(x) = 0
Factor sin(x) from the left hand side:
sin(x) (sec(x) - 1) = 0
Split sin(x) (sec(x) - 1) into separate parts with additional assumptions.
Assume cos(x)!=0 from sec(x):
sec(x) - 1 = 0 or sin(x) = 0 for cos(x)!=0
Add 1 to both sides:
sec(x) = 1 or sin(x) = 0 for cos(x)!=0
Take the reciprocal of both sides:
cos(x) = 1 or sin(x) = 0 for cos(x)!=0
Take the inverse cosine of both sides:
x = 2 π n_1 for n_1 element Z
or sin(x) = 0 for cos(x)!=0
Take the inverse sine of both sides:
x = 2 π n_1 for n_1 element Z
or x = π n_2 for cos(x)!=0 and n_2 element Z
The roots x = π n_2 never violate cos(x)!=0, which means this assumption can be omitted:
Answer: x = 2 π n_1 for n_1 element Z
or x = π n_2 for n_2 element Z