Question

What is the number of solutions of tanx + secx = 2 cosx if x belongs to (0, 2π)?

Answer 1

To find the number of solutions of the equation tan(x) + sec(x) = 2cos(x) in the interval (0, 2π), we can analyze the behavior of the individual functions involved.

Rewriting the equation using trigonometric identities, we have: sin(x)/cos(x) + 1/cos(x) = 2cos(x)

Combining the fractions on the left-hand side, we get: (sin(x) + 1)/cos(x) = 2cos(x)

Multiplying both sides by cos(x) (assuming cos(x) ≠ 0): sin(x) + 1 = 2cos^2(x)

Using the identity sin^2(x) + cos^2(x) = 1, we substitute sin^2(x) = 1 - cos^2(x): 1 - cos^2(x) + 1 = 2cos^2(x)

Rearranging the equation, we have: 3cos^2(x) - 1 = 0

Simplifying further, we get: cos^2(x) = 1/3

Taking the square root of both sides, we have: cos(x) = ±1/√3

From the interval (0, 2π), we can see that the cosine function is positive in the first and fourth quadrants. So, we only consider the positive solution: cos(x) = 1/√3

Using the inverse cosine function, we have: x = cos^(-1)(1/√3)

Evaluating the inverse cosine, we find: x = π/6

Thus, we have found a single solution for the given equation in the interval (0, 2π).