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 expression.

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

3. Set each factor to zero using the zero product property to find the values of x that satisfy the identity.

cos(x)=0

2*sin(x)+1=0

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

x=π/2+n*π

5. Solve for x in the second equation by isolating the sine function.

sin(x)=−1/2

x=(7*π)/6+2*n*π

x=(11*π)/6+2*n*π

Final Answer

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

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