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

Problem

cos(2*x)−sin(x)=1

Solution

1. Apply the double angle identity for cos(2*x) to rewrite the equation in terms of sin(x) using cos(2*x)=1−2*sin2(x)

1−2*sin2(x)−sin(x)=1

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

−2*sin2(x)−sin(x)=0

3. Multiply by -1 to make the leading coefficient positive.

2*sin2(x)+sin(x)=0

4. Factor out the common term sin(x)

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

5. Set each factor to zero to find the possible values for sin(x)

sin(x)=0

2*sin(x)+1=0⇒sin(x)=−1/2

6. Solve for x by finding the angles that satisfy these equations within the general solution.
   For sin(x)=0

x=n*π

For sin(x)=−1/2

x=(7*π)/6+2*n*π

x=(11*π)/6+2*n*π

Final Answer

cos(2*x)−sin(x)=1⇒x=n*π,(7*π)/6+2*n*π,(11*π)/6+2*n*π

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