Final answer:
The expression (cos2x + 2cosx + 1) / (cosx + 1) simplifies to cosx + 1 after recognizing the numerator as a perfect square trinomial and canceling out common factors.
Step-by-step explanation:
To simplify the expression (cos2x + 2cosx + 1) / (cosx + 1), we can look at the numerator and notice that it is a perfect square trinomial. The expression cos2x + 2cosx + 1 can be factored into (cosx + 1)2. When we place this over the denominator, we end up with (cosx + 1)2 / (cosx + 1).
This simplifies further because we can cancel out one (cosx + 1) from the numerator with the (cosx + 1) in the denominator. So, our final simplified expression is simply cosx + 1, since (cosx + 1) / (cosx + 1) equals 1. It is important to note that cosx must not equal -1 since this would make the denominator zero, which is undefined.