Solve for x ( square root of 2)/2*csc(x)-1=0

Problem

√(,2)/2*csc(x)−1=0

Solution

1. Isolate the cosecant term by adding 1 to both sides of the equation.

√(,2)/2*csc(x)=1

2. Solve for csc(x) by multiplying both sides by the reciprocal 2/√(,2)

csc(x)=2/√(,2)

3. Simplify the expression by rationalizing the denominator or recognizing that 2/√(,2)=√(,2)

csc(x)=√(,2)

4. Rewrite the equation in terms of the sine function using the identity csc(x)=1/sin(x)

1/sin(x)=√(,2)

5. Invert both sides to find the value of sin(x)

sin(x)=1/√(,2)

6. Rationalize the denominator to identify the standard unit circle value.

sin(x)=√(,2)/2

7. Determine the general solutions for x where the sine value is √(,2)/2 which occurs at π/4 and (3*π)/4 in the first rotation.

x=π/4+2*n*π

x=(3*π)/4+2*n*π

Final Answer

√(,2)/2*csc(x)−1=0⇒x=π/4+2*n*π,(3*π)/4+2*n*π

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