Expand the Trigonometric Expression cos(x)^4

Problem

cos(x)

Solution

1. Apply the power-reduction formula for cos(x) Recall that cos(x)=(1+cos(2*x))/2

2. Rewrite the expression as the square of a square.

cos(x)=(cos(x))2

3. Substitute the formula into the expression.

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

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

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

5. Apply the power-reduction formula again to the cos(2*x) term. Note that cos(2*x)=(1+cos(4*x))/2

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

6. Simplify the fraction by multiplying the numerator and denominator by 2 to clear the internal fraction.

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

7. Combine like terms in the numerator.

cos(x)=(3+4*cos(2*x)+cos(4*x))/8

Final Answer

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

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