Solve for x cos(2x)- square root of 2sin(x)=1

Problem

cos(2*x)−√(,2)*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)−√(,2)*sin(x)=1

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

−2*sin2(x)−√(,2)*sin(x)=0

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

2*sin2(x)+√(,2)*sin(x)=0

4. Factor out the common term sin(x)

sin(x)*(2*sin(x)+√(,2))=0

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

sin(x)=0

2*sin(x)+√(,2)=0

6. Solve for x in the first case sin(x)=0

x=n*π

7. Solve for x in the second case 2*sin(x)=−√(,2) which simplifies to sin(x)=−√(,2)/2

x=(5*π)/4+2*n*π

x=(7*π)/4+2*n*π

Final Answer

cos(2*x)−√(,2)*sin(x)=1⇒x=n*π,(5*π)/4+2*n*π,(7*π)/4+2*n*π

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