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,....**