Solve for x cos(x)^2+2cos(x)+1=0

Problem

cos(x)+2*cos(x)+1=0

Solution

1. Identify the equation as a quadratic in terms of cos(x)

2. Factor the quadratic expression on the left side, noting it is a perfect square trinomial of the form a2+2*a*b+b2=(a+b)2

(cos(x)+1)2=0

3. Solve for cos(x) by taking the square root of both sides.

cos(x)+1=0

cos(x)=−1

4. Determine the values of x that satisfy the equation. For cos(x)=−1 the general solution involves odd multiples of π

x=π+2*n*π

5. Simplify the expression for x where n is any integer.

x=(2*n+1)*π

Final Answer

cos(x)+2*cos(x)+1=0⇒x=(2*n+1)*π

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