Expand the Trigonometric Expression sin(x)^4

Problem

sin^4(x)

Solution

1. Apply the power-reduction formula for sin^2(x) Use the identity sin^2(x)=(1−cos(2*x))/2 and square the entire expression.

sin^4(x)=(sin^2(x))^2

sin^4(x)=((1−cos(2*x))/2)^2

2. Expand the squared binomial in the numerator. Square the denominator as well.

sin^4(x)=(1−2*cos(2*x)+cos^2(2*x))/4

3. Apply the power-reduction formula again for the cos^2(2*x) term. Use the identity cos^2(θ)=(1+cos(2*θ))/2 where θ=2*x

cos^2(2*x)=(1+cos(4*x))/2

4. Substitute the identity back into the expression and simplify the fractions.

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

5. Distribute the constant and combine like terms to reach the final expanded form.

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

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

sin^4(x)=3/8−1/2*cos(2*x)+1/8*cos(4*x)

Final Answer

sin^4(x)=3/8−1/2*cos(2*x)+1/8*cos(4*x)

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