cos x+sec x=2
if you could please show all work
Certainly! To solve the equation cos x + sec x = 2, we can utilize the trigonometric identity sec x = 1/cos x.
Substituting this identity into the equation, we have cos x + 1/cos x = 2.
To eliminate the denominator, we can multiply the entire equation by cos x, giving us cos^2 x + 1 = 2cos x.
Rearranging the terms, we get cos^2 x - 2cos x + 1 = 0.
This quadratic equation can be factored as (cos x - 1)(cos x - 1) = 0, or (cos x - 1)^2 = 0.
Taking the square root of both sides, we have cos x - 1 = 0, or cos x = 1.
To find the values of x that satisfy cos x = 1, we can refer to the unit circle. On the unit circle, the cosine function has a value of 1 at x = 0 and x = 2π.
Therefore, the solutions to the equation cos x + sec x = 2 are x = 0 and x = 2π.
You can verify this by substituting these values back into the original equation.