Question

Find all solutions to the equation.

cos2x + 2 cos x + 1 = 0

Answer 1

first of all, we must use the equivalent formula of cos2x, it is cos2x=2cos^2x-1,
the equation has become 2cos^2x-1 +2cosx +1 = 0, it means 2cos^2x+2cosx = 0, or
cosx(cosx+1)=0, we get cosx=0, implying x=Pi/2, or (cosx+1)=0,  cosx= -1
implying x = Pi
for x=0, cos2*0+2cos0+1=1+2+1=4 impossible
for x= Pi, cos2*Pi+2cosPi+1=1-2+1=0 it is exact , so the solution is x=Pi

Answer 2

Answer:

Here, the given expression is,

$cos(2x)+2cosx+1=0$

$\implies (2cos^2x-1)+2cosx+1=0$  ( Because cos(2x) = 2cos²x-1 )

$\implies 2cos^2x+2cosx=0$

$\implies 2cosx(cosx+1)=0$

Thus, 2 cos x = 0 or cos x + 1 = 0

If ,

$2cos x = 0$

⇒ $cos x = 0$

⇒ $x = \frac{pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}.......$

⇒ $x = \frac{\pi}{2}+n\pi$

Where, n = 0, 1, 2, ......

Now, if,

$cos x = -1$

⇒ $x=\pi, 3\pi, 5\pi,.....$

⇒ $x = \pi+n\pi$

Where, n = 0, 1, 2, ......