Simplify 19sin(x)^4

Problem

19*sin4(x)

Solution

1. Identify the expression as a power of a trigonometric function, specifically 19*(sin2(x))2

2. Apply the power-reduction formula for sine, which states sin2(x)=(1−cos(2*x))/2

3. Substitute the formula into the expression:

19*((1−cos(2*x))/2)2

4. Expand the squared binomial in the numerator and square the denominator:

19/4*(1−2*cos(2*x)+cos2(2*x))

5. Apply the power-reduction formula again for cos2(2*x) which states cos2(2*x)=(1+cos(4*x))/2

6. Substitute and distribute the terms:

19/4*(1−2*cos(2*x)+(1+cos(4*x))/2)

7. Combine the constant terms 1+1/2=3/2 inside the parentheses:

19/4*(3/2−2*cos(2*x)+cos(4*x)/2)

8. Distribute the fraction 19/4 to reach the final simplified form:

57/8−(19*cos(2*x))/2+(19*cos(4*x))/8

Final Answer

19*sin4(x)=(57−76*cos(2*x)+19*cos(4*x))/8

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