ctgx+cosx = 1+ctgx*cosx find x
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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,....
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