Solve for x cot(x/2)=0

Problem

cot(x/2)=0

Solution

1. Identify the definition of the cotangent function in terms of cosine and sine.

cot(x/2)=cos(x/2)/sin(x/2)

2. Set the numerator equal to zero, as a fraction is zero only when its numerator is zero and its denominator is non-zero.

cos(x/2)=0

3. Determine the values where the cosine function equals zero.

x/2=π/2+n*π

where n is any integer (n∈ℤ.

4. Solve for x by multiplying both sides of the equation by 2.

x=2*(π/2+n*π)

5. Simplify the expression to find the general solution.

x=π+2*n*π

6. Verify that these values do not make the denominator sin(x/2) equal to zero. Since sin(π/2+n*π)=±1 the expression is defined for all these values.

Final Answer

cot(x/2)=0⇒x=π+2*n*π,n∈ℤ

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