I keep trying to find the power reducing formula for sin^4(x), but I can't seem to get all the fractional parts correct.
The answer I should be getting is:
sin^4(x)=(1/8)cos4x-(1/2)cos2x+(3/8)
I can only get this far knowing feeling confident. When I go further I start making mathematical errors.
((1-cos2x)/2)*((1-cos2x)/2)
To find the power reducing formula for sin^4(x), we can make use of the double angle formula for cosine. Here's the step-by-step explanation:
1. Start with the expression sin^4(x).
2. Expand it using the binomial theorem (also known as FOIL or the distributive property) as follows:
sin^4(x) = (sin^2(x))^2
= (1 - cos^2(x))^2
3. Apply the difference of squares formula to (1 - cos^2(x))^2:
sin^4(x) = (1 - cos^2(x))^2
= (1 - cos^2(x))(1 - cos^2(x))
4. Now, distribute the terms:
sin^4(x) = (1 - 2cos^2(x) + cos^4(x))
5. We can simplify the expression further by using the double angle formula for cosine:
cos(2x) = 2cos^2(x) - 1
Rearranging the equation:
2cos^2(x) = cos(2x) + 1
Substituting this into the expression:
sin^4(x) = (1 - (cos(2x) + 1) + cos^4(x))
= (1 - cos(2x) - 1 + cos^4(x))
= -cos(2x) + cos^4(x)
6. However, the answer you're looking for is in a different form:
sin^4(x) = (1/8)cos(4x) - (1/2)cos(2x) + (3/8)
To isolate the terms and compare, we need to simplify (-cos(2x) + cos^4(x)).
7. To do this, multiply all terms by a common denominator of 8:
sin^4(x) = -8/8 * cos(2x) + 8/8 * cos^4(x)
This gives:
sin^4(x) = (-8cos(2x) + 8cos^4(x))/8
8. Simplify further by dividing through by 8:
sin^4(x) = -cos(2x) + cos^4(x)
Now we can see that the expression (-cos(2x) + cos^4(x)) matches the desired form.
So, the correct power reducing formula for sin^4(x) is indeed:
sin^4(x) = (1/8)cos(4x) - (1/2)cos(2x) + (3/8)
I hope this clears up any confusion and helps you get the correct answer.