Question

COS²x + COSx = -Sin²x...

Answer 1

COS²x + COSx = -Sin²x
( cos²x+sin²x)+cosx=0
cosx+1=0
cosx=-1
x=π+2πN

Answer 2

Cos²x+cosx+sin²x=0
cosx+1=0
cosx=-1
x=π+2πn

Answer 3

There are different ways to approach this trigonometric identity, but one common method is to use the Pythagorean identity:

sin²x + cos²x = 1

From this identity, we can derive:

sin²x = 1 - cos²x

Substituting this into the given equation:

cos²x + cosx = -sin²x

cos²x + cosx = - (1 - cos²x)

Simplifying:

2cos²x + cosx + 1 = 0

This is now a quadratic equation in terms of cosx. We can solve for cosx using the quadratic formula:

cosx = (-b ± sqrt(b² - 4ac)) / 2a

where a = 2, b = 1, and c = 1. Substituting these values:

cosx = (-1 ± sqrt(1 - 8)) / 4

cosx = (-1 ± sqrt(-7)) / 4

cosx = (-1 ± i√7) / 4

where i is the imaginary unit. Therefore, the solutions for cosx are complex numbers.

In conclusion, the trigonometric identity does not hold for real values of x.