Solve for x 2sin(2x)sin(x)-cos(x)=0

Problem

2*sin(2*x)*sin(x)−cos(x)=0

Solution

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

2*(2*sin(x)*cos(x))*sin(x)−cos(x)=0

2. Simplify the expression by multiplying the terms.

4*sin2(x)*cos(x)−cos(x)=0

3. Factor out the common term cos(x)

cos(x)*(4*sin2(x)−1)=0

4. Set each factor to zero to find the possible solutions for x

cos(x)=0

4*sin2(x)−1=0

5. Solve the first equation cos(x)=0

x=π/2+n*π

6. Solve the second equation 4*sin2(x)−1=0 by isolating sin(x)

sin2(x)=1/4

sin(x)=±1/2

7. Find the values of x for sin(x)=1/2 and sin(x)=−1/2

x=π/6+n*π

x=(5*π)/6+n*π

8. Combine the solutions into a general form where n is any integer.

Final Answer

x=π/2+n*π,π/6+n*π,(5*π)/6+n*π

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