Final answer:
To solve the equation sec2x + 2secx = 0, we can factor out a common term of secx and solve each equation separately. The solutions to the equation are x = 2π/3 + 2πk and x = 4π/3 + 2πk, where k is an integer.
Step-by-step explanation:
To solve the equation sec2x + 2secx = 0, we can first factor out a common term of secx:
secx(secx + 2) = 0
So, either secx = 0 or secx + 2 = 0. We can solve each equation separately:
1. secx = 0:
Since secx is the reciprocal of cosx, the equation secx = 0 means that cosx = 1/0, which is not defined. So, there are no real solutions for this part of the equation.
2. secx + 2 = 0:
Subtracting 2 from both sides, we get secx = -2. Since secx is the reciprocal of cosx, the equation secx = -2 means that cosx = -1/2. We can find the solutions for cosx = -1/2 by referring to the unit circle or using the inverse cosine function.
From the unit circle, we can determine that the solutions are x = 2π/3 + 2πk and x = 4π/3 + 2πk, where k is an integer.
So, the solutions to the equation are x = 2π/3 + 2πk and x = 4π/3 + 2πk, where k is an integer.