The solution set of the trigonometric equation
cos2x−2cosx=−1
for 0≤x≤2π is ....
Explanation
Given the equation cos2x−2cosx=−1 for 0≤x≤2π
Transforming the Equation
Use the trigonometric identity cos2x=2cos2x−1
cos2x−2cosx=−1
(2cos2x−1)−2cosx+1=0
2cos2x−2cosx=0
cosx(cosx−1)=0
Therefore cosx=0 or cosx=1
Solving for cos x = 0
If cosx=cosα, then
x1=α+k⋅2πandx2=−α+k⋅2π
For cosx=cos2π=0
k=0⇒x1=2π+0⋅2π=21π
k=1⇒x2=−2π+2π=23π
Solving for cos x = 1
For cosx=cos0=1
k=0⇒x1=0+0⋅2π=0
k=1⇒x2=−0+2π=2π
Solution Set
Therefore, the solution set of the equation cos2x−2cosx=−1 for 0≤x≤2π is
{0,21π,23π,2π}