ctgx+cosx = 1+ctgx*cosx find x

ctgx + cosx = 1+
ctg*cosx

First we will rewrite the
identities.

We know
that:

ctg(x) =
cosx/sinx.

==> cosx/sinx + cosx = 1 + cosx/sinx *
cosx

Now we will
simplify.

==> (cosx + cosx*sinx)/sinx = 1+ 
1/sinx

==> We will multiply by
sinx.

==> cosx + cosx*sinx = sinx +
1

Now we will move all terms to the left
side.

==> cosx + cosx*sinx - sinx -1 =
0

Now we will facotr cosx and -1
.

==> cosx ( 1+ sinx ) -1 ( sinx+ 1) =
0

Now we will factor
(1+sinx)

==> (1+sinx) (cosx -1) =
0

==> sinx +1 = 0 ==> sinx = -1  ==> x
= 3pi/2 + 2npi

==> cosx -1 = 0==> cosx = 1
==> x = 0+2npi, pi+2npi, 2pi+2npi

Then the answer
is:

x = { 2npi, 3pi/2+2npi }  n= 0, 1, 2,
3,....