Verify the Identity sin(x)+sin(5x)=tan(3x)

Problem

sin(x)+sin(5*x)=2*sin(3*x)*cos(2*x)

Solution

1. Identify the sum-to-product formula for sine, which states sin(A)+sin(B)=2*sin((A+B)/2)*cos((A−B)/2)

2. Substitute the values A=5*x and B=x into the formula to simplify the left side of the expression.

3. Calculate the arguments for the sine and cosine functions by performing the addition and subtraction.

(5*x+x)/2=(6*x)/2=3*x

(5*x−x)/2=(4*x)/2=2*x

4. Combine the results to rewrite the sum as a product.

sin(5*x)+sin(x)=2*sin(3*x)*cos(2*x)

Final Answer

sin(x)+sin(5*x)=2*sin(3*x)*cos(2*x)

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