Solve the equation if x is in (0,2pi) cos^2x+cosx-6=0

We'll solve the trigonometric equation using substitution
technique.

Let cos x = t

t^2 +
t - 6 = 0

We'll apply quadratic
formula:

t1 = [-1+sqrt(1 +
24)]/2

t1 = (-1 + 5)/2

t1 =
2

t2 = (-1-5)/2

t2 =
-3

But cos x = t

cos x = t1
<=> cos x = 2 impossible because the values of cosine function cannot be
larger than 1.

cos x = t2 <=> cos x = -3
impossible because the values of cosine function cannot be smaller than
-1.

Therefore, the given equation has no
solution over the interval (0 , 2`pi` ).