Verify the Identity sin(2x)-cos(x)=0

Problem

sin(2*x)−cos(x)=0

Solution

1. Apply the double angle identity for sine, which states sin(2*x)=2*sin(x)*cos(x)

2*sin(x)*cos(x)−cos(x)=0

2. Factor out the common term cos(x) from the left side of the equation.

cos(x)*(2*sin(x)−1)=0

3. Set each factor to zero using the zero product property to find the solutions for x

cos(x)=0

2*sin(x)−1=0

4. Solve for x in the first equation where the cosine is zero.

x=π/2+n*π

5. Isolate the sine function in the second equation.

2*sin(x)=1

sin(x)=1/2

6. Solve for x in the second equation where the sine is 1/2

x=π/6+2*n*π

x=(5*π)/6+2*n*π

Final Answer

sin(2*x)−cos(x)=cos(x)*(2*sin(x)−1)=0

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