Find the number of solutions of tan x + sec x = 2 cos x in [0, 2π).

To find the number of solutions of the equation tan x + sec x = 2 cos x in the interval [0, 2π), we can follow these steps:

Step 1: Simplify the equation using trigonometric identities. Using the identity sec x = 1/cos x, we can rewrite the equation as tan x + 1/cos x = 2 cos x.

Step 2: Multiply both sides of the equation by cos x to eliminate the denominators. This gives us cos x * tan x + 1 = 2 cos^2 x.

Step 3: Rewrite tan x in terms of sin x and cos x. Using the identity tan x = sin x / cos x, we can rewrite the equation as sin x + cos x = 2 cos^2 x.

Step 4: Rearrange the equation to get it in a quadratic form. Move all the terms to one side of the equation to obtain 2 cos^2 x - sin x - cos x = 0.

Step 5: Apply the quadratic formula to solve for cos x. Using the quadratic formula, cos x = (-b ± √(b^2 - 4ac)) / (2a), where a = 2, b = -1, and c = -1.

Step 6: Simplify the expression under the square root. Calculating b^2 - 4ac, we get (-1)^2 - 4(2)(-1) = 1 + 8 = 9.

Step 7: Solve for cos x. Applying the quadratic formula, we have cos x = (-(-1) ± √9) / (2(2)). Simplifying further, cos x = (1 ± 3) / 4.

Step 8: Find the values of cos x. There are two possible solutions for cos x: cos x = (1 + 3) / 4 = 4/4 = 1, and cos x = (1 - 3) / 4 = -2/4 = -1/2.

Step 9: Find the corresponding values of x. To find the values of x, we can use the inverse cosine function. For cos x = 1, x = cos^(-1)(1) = 0. For cos x = -1/2, x = cos^(-1)(-1/2) = 2π/3.

Step 10: Check if the solutions are within the given interval. The solutions x = 0 and x = 2π/3 are both within the interval [0, 2π).

Step 11: Determine the number of solutions. Since we have found two solutions that satisfy the equation within the given interval, the number of solutions is 2.

Therefore, the equation tan x + sec x = 2 cos x has 2 solutions in the interval [0, 2π).