Solve for x 2sin(x)+csc(x)=0

Problem

2*sin(x)+csc(x)=0

Solution

1. Use the reciprocal identity to rewrite the cosecant function in terms of sine.

csc(x)=1/sin(x)

2. Substitute the identity into the original equation.

2*sin(x)+1/sin(x)=0

3. Multiply the entire equation by sin(x) to eliminate the fraction, noting that sin(x)≠0

2*sin2(x)+1=0

4. Isolate the squared sine term by subtracting 1 and dividing by 2.

2*sin2(x)=−1

sin2(x)=−1/2

5. Analyze the resulting equation for real solutions. Since the square of any real number must be non-negative, sin2(x) cannot equal a negative value.

sin2(x)≥0

Final Answer

2*sin(x)+csc(x)=0⇒No real solutions

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