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

Problem

cos(x)−sin(x)=1

Solution

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

1/√(,2)*cos(x)−1/√(,2)*sin(x)=1/√(,2)

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

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

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

sin(π/4−x)=√(,2)/2

4. Solve for the argument by identifying where the sine function equals √(,2)/2

π/4−x=π/4+2*n*π

π/4−x=(3*π)/4+2*n*π

5. Isolate x in both cases, where n is any integer.

x=−2*n*π

x=−π/2−2*n*π

6. Simplify the general solution notation.

x=2*k*π

x=(3*π)/2+2*k*π

Final Answer

cos(x)−sin(x)=1⇒x=2*k*π,(3*π)/2+2*k*π

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