Simplify sin(x)^4

Problem

sin4(x)

Solution

1. Apply the power-reduction identity for sin2(x) which is sin2(x)=(1−cos(2*x))/2

2. Rewrite the expression as the square of sin2(x)

sin4(x)=(sin2(x))2

3. Substitute the identity into the expression.

sin4(x)=((1−cos(2*x))/2)2

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

sin4(x)=(1−2*cos(2*x)+cos2(2*x))/4

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

sin4(x)=1/4*(1−2*cos(2*x)+(1+cos(4*x))/2)

6. Distribute the fraction and combine like terms to reach the final simplified form.

sin4(x)=1/4−1/2*cos(2*x)+1/8+1/8*cos(4*x)

sin4(x)=3/8−1/2*cos(2*x)+1/8*cos(4*x)

Final Answer

sin4(x)=(3−4*cos(2*x)+cos(4*x))/8

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