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

Problem

sin(x)−cos(x)=1

Solution

1. Divide both sides of the equation by √(,2) to prepare for a trigonometric identity substitution.

1/√(,2)*sin(x)−1/√(,2)*cos(x)=1/√(,2)

2. Substitute the values cos(π/4)=1/√(,2) and sin(π/4)=1/√(,2) into the equation.

sin(x)*cos(π/4)−cos(x)*sin(π/4)=1/√(,2)

3. Apply the sine difference identity sin(A−B)=sin(A)*cos(B)−cos(A)*sin(B)

sin(x−π/4)=1/√(,2)

4. Determine the general solutions for the angle by identifying where the sine function equals 1/√(,2)

x−π/4=π/4+2*n*π

x−π/4=(3*π)/4+2*n*π

5. Isolate x by adding π/4 to both sides of each equation, where n is any integer.

x=π/2+2*n*π

x=π+2*n*π

Final Answer

sin(x)−cos(x)=1⇒x=π/2+2*n*π,π+2*n*π

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