Verify the Identity csc(x)(2sin(x)- square root of 2)=0

Problem

csc(x)*(2*sin(x)−√(,2))=0

Solution

1. Identify the domain of the equation by noting that csc(x)=1/sin(x) which implies sin(x)≠0

2. Apply the zero product property to the factors of the equation.

3. Analyze the first factor csc(x)=0 Since csc(x)=1/sin(x) and a fraction is only zero if its numerator is zero, csc(x) can never be zero.

4. Solve the second factor by setting 2*sin(x)−√(,2)=0

5. Isolate the sine function by adding √(,2) to both sides and dividing by 2

2*sin(x)=√(,2)

sin(x)=√(,2)/2

6. Determine the values of x within one period [0,2*π) where the sine is √(,2)/2

x=π/4

x=(3*π)/4

7. Generalize the solution for all real numbers by adding multiples of the period 2*π

x=π/4+2*k*π

x=(3*π)/4+2*k*π

Final Answer

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

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