Solve tan x=sin x giving all possible solutions

x = 2 π n_1 for n_1 element Z

or x = π n_2 for n_2 element Z

Step-by-step 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