Simplify cos(x)^2+2cos(x)+1

Problem

cos^2(x)+2*cos(x)+1

Solution

1. Identify the structure of the expression as a quadratic form u^2+2*u+1 where u=cos(x)

2. Recognize the perfect square trinomial pattern a^2+2*a*b+b^2=(a+b)^2

3. Substitute a=cos(x) and b=1 into the pattern.

4. Rewrite the expression in its factored, simplified form.

Final Answer

cos^2(x)+2*cos(x)+1=(cos(x)+1)^2

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