Question

Solve tan x=sin x giving all possible solutions​

Answer 1

Answer:

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