The equation we want to solve is:
cos(2x) + 2 cos(x) + 1 = 0
First, we can use the double angle identity for cosine, which states:
cos(2x) = 2cos²(x) – 1
Substituting this into our equation gives:
2cos²(x) – 1 + 2cos(x) + 1 = 0
Simplifying this, we have:
2cos²(x) + 2cos(x) = 0
Next, we can factor out a common term:
2cos(x)(cos(x) + 1) = 0
This gives us two cases to solve:
- 2cos(x) = 0
- cos(x) + 1 = 0
Let’s solve each case:
- For the first case, 2cos(x) = 0:
- For the second case, cos(x) + 1 = 0:
This simplifies to cos(x) = 0. The general solutions for this are:
x = (2n + 1) * π/2, where n is any integer.
This simplifies to cos(x) = -1. The general solution for this is:
x = (2n + 1) * π, where n is any integer.
In conclusion, combining the solutions from both cases, we find:
The solutions to the equation cos(2x) + 2cos(x) + 1 = 0 are:
x = (2n + 1) * π/2 and x = (2n + 1) * π, where n is any integer.