cos^2x-sin^2 x+1=cos x
sin^2 x = 1 - cos^2 x
so we have
cos^2 x - 1 + cos^2 x + 1
or
2 cos^2 x
on the left
so
2 cos^2 x = cos x
cos x = 1/2
x = 60 degrees or pi/3 radians
or
300 degrees or 5 pi/3 radians
from Damon's middle section:
2cos^2 x - cosx = 0
cosx(2cosx - 1) = 0
cosx = 0 or cosx = 1/2
so from cosx=1/2 we have the 2 answers that Damon gave
from cosx = 0 , we also have
x = 90° or x = 270°
x = π/2 or x = 3π/2
To solve the equation cos^2x - sin^2 x + 1 = cos x, we can start by using a trigonometric identity. The identity we can use here is cos^2x - sin^2 x = cos(2x).
Substituting this identity into the equation, we get:
cos(2x) + 1 = cos x
Now, we can simplify the equation further:
cos(2x) = cos x - 1
We can now solve for x by considering the possible values of cos(2x) and cos x - 1.
1. If cos(2x) = cos x - 1, we can solve it algebraically:
cos(2x) - cos x = -1
By applying the double-angle formula for cosine, we get:
2cos^2 x - cos x - 1 = 0
Now, we can make a substitution to simplify the quadratic equation:
Let z = cos x
Substituting this, we have:
2z^2 - z - 1 = 0
We can now factor this equation:
(2z + 1)(z - 1) = 0
Solving for z gives us:
z = -1/2 or z = 1
Since z represents cos x, we have two possible solutions:
cos x = -1/2 or cos x = 1
To find the values of x, we can use the inverse cosine function:
x = arccos(-1/2) or x = arccos(1)
Using a unit circle or trigonometric table, we can find that:
x = 120 degrees + n*360 degrees or x = 0 degrees + n*360 degrees, where n is an integer.
2. Another possibility is when cos(2x) = -(cos x - 1):
cos(2x) + cos x - 1 = 0
To solve this equation, we can use the quadratic formula:
z = (-b ± √(b^2 - 4ac)) / 2a
In this case, let z = cos x:
cos x = (-1 ± √(1 + 4*1*1)) / 2
cos x = (-1 ± √(5)) / 2
Using the inverse cosine function to find the values of x:
x = arccos((-1 + √5) / 2) or x = arccos((-1 - √5) / 2)
Again, using a unit circle or trigonometric table, we can find that:
x = 72 degrees or x = 288 degrees
So, the solution to the given equation is:
x = 0 degrees + n*360 degrees, 72 degrees + n*360 degrees, 120 degrees + n*360 degrees, or 288 degrees + n*360 degrees, where n is an integer.