Solve for x sin(x)^2-cos(x)^2=1+cos(x)

Problem

sin(x)−cos(x)=1+cos(x)

Solution

1. Apply the Pythagorean identity to replace sin(x) with 1−cos(x) so the equation is in terms of a single trigonometric function.

1−cos(x)−cos(x)=1+cos(x)

2. Combine like terms on the left side of the equation.

1−2*cos(x)=1+cos(x)

3. Subtract 1 from both sides to simplify the equation.

−2*cos(x)=cos(x)

4. Rearrange the equation into standard quadratic form by adding 2*cos(x) to both sides.

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

5. Factor out the common term cos(x)

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

6. Set each factor to zero to find the possible values for cos(x)

cos(x)=0

2*cos(x)+1=0⇒cos(x)=−1/2

7. Solve for x using the unit circle values within the standard interval [0,2*π)
   For cos(x)=0

x=π/2,(3*π)/2

For cos(x)=−1/2

x=(2*π)/3,(4*π)/3

8. Generalize the solution by adding 2*π*n for any integer n

Final Answer

sin(x)−cos(x)=1+cos(x)⇒x=π/2+π*n,(2*π)/3+2*π*n,(4*π)/3+2*π*n

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