Verify the Identity sin(3x)=3sin(x)-4sin(x)^3

Problem

sin(3*x)=3*sin(x)−4*sin3(x)

Solution

1. Rewrite the argument 3*x as the sum of 2*x and x to prepare for the angle addition formula.

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

2. Apply the sine addition formula sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

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

3. Substitute the double angle identities sin(2*x)=2*sin(x)*cos(x) and cos(2*x)=1−2*sin2(x) into the expression.

sin(3*x)=(2*sin(x)*cos(x))*cos(x)+(1−2*sin2(x))*sin(x)

4. Simplify the terms by multiplying the trigonometric functions.

sin(3*x)=2*sin(x)*cos2(x)+sin(x)−2*sin3(x)

5. Use the Pythagorean identity cos2(x)=1−sin2(x) to express the entire equation in terms of sine.

sin(3*x)=2*sin(x)*(1−sin2(x))+sin(x)−2*sin3(x)

6. Distribute the 2*sin(x) through the parentheses.

sin(3*x)=2*sin(x)−2*sin3(x)+sin(x)−2*sin3(x)

7. Combine like terms to reach the final form of the identity.

sin(3*x)=3*sin(x)−4*sin3(x)

Final Answer

sin(3*x)=3*sin(x)−4*sin3(x)

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