# Solve the equation tan x = sin x if x is in ( 0, 2pi ).

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tanx = sinx is the given equation to solve for x:

Solution:

Tanx = sinx/cosx, by definition. Substitute this in the given equation.

So the give equation becomes:

sinx/cosx = sinx.

Divide both sides by sinx.

1/cosx = 1.

Cross multiply.

1 = cosx

cosx = 1 when x = 0 or 2npi, n = 0,1,2....

Therefore,

x = 2npi, n =0 , 1 ,2 ,3, ....

First, we'll write the tangent function as a ratio:

tan x = sin x / cos x

We'll re-write the equation:

sin x / cos x = sin x

We'll cross multiply:

sin x = sin x*cos x

We'll subtract the product sin x*cos x:

sin x - sin x*cos x = 0

We'll factorize by sin x:

sin x(1 - cos x) = 0

We'll put each factor as 0:

sin x = 0

x = 0

or

x = pi

1 - cos x = 0

cos x = 1

x = 0

**But we notice that x belongs to the interval (0,2pi) so x = 0 is not a valid solution, since 0 doesn't belong to the interval. The only valid solution of the equation is x = pi.**