MATH HELP PLEASE?

Question

cos^2x+cos2x=-1/2

also this one:

sin^2x + sinx + 1 = 0

Answer 1

cos²x + cos2x = -½

Substitute : cos2x = 2cos²x - 1

cos²x + 2cos²x - 1 = -½

3cos²x = 1-½

3cos²x = ½

cos²x = 1/6

cosx = √(1/6) = 1/√6

cosx = cos(65.9°)

Solution : x = 2nπ ± 65.9°  where x is integer.

Answer 2

sin²x + sinx + 1 = 0

(sinx)² + sinx + 1 = 0

Using formula for roots of quadratic equation : [- b ± √(b²-4ac)]/2a

a = 1 , b = 1 , c = 1

sinx = [-1 ± √(1²-4*1*1)]/2*1 = [-1 ± √(-3)]/2 = [-1 ± i√3]/2

Here sinx value is complex/imaginary value.

So roots does not exist.

Solution :  Equation sin²x + sinx + 1 = 0 has no solutions.