Rewrite the expression in terms of the first power of the cosine
sin^8(x)
how do i do this plz help !!
To rewrite sin^8(x) in terms of the first power of cosine, you can use the Pythagorean Identity which states that sin^2(x) + cos^2(x) = 1. By rearranging this equation, we can express sin^2(x) in terms of cos^2(x): sin^2(x) = 1 - cos^2(x).
To rewrite sin^8(x) using this identity, we can substitute sin^2(x) with (1 - cos^2(x)):
sin^8(x) = (1 - cos^2(x))^4.
Now, we can expand the expression using the binomial theorem:
sin^8(x) = (1 - cos^2(x))^4 = 1 - 4cos^2(x) + 6cos^4(x) - 4cos^6(x) + cos^8(x).
Therefore, sin^8(x) can be rewritten in terms of the first power of cosine as:
sin^8(x) = 1 - 4cos^2(x) + 6cos^4(x) - 4cos^6(x) + cos^8(x).
To rewrite the expression sin^8(x) in terms of the first power of cosine, you can use the trigonometric identity known as the Pythagorean Identity. The Pythagorean Identity states that sin^2(x) + cos^2(x) = 1.
1. First, let's rewrite sin^8(x) as (sin^2(x))^4. This is because sin^8(x) is equivalent to (sin^2(x))^4.
2. Now, substitute sin^2(x) in terms of cos^2(x) using the Pythagorean Identity. Since sin^2(x) + cos^2(x) = 1, we can rearrange it as sin^2(x) = 1 - cos^2(x).
3. Substitute the expression for sin^2(x) from step 2 into sin^8(x).
(sin^2(x))^4 = (1 - cos^2(x))^4
This expression is now written in terms of the first power of cosine.
I can get it in powers of cos but not first power
sin^8 = sin^2 * sin^2 *sin^2 *sin^2
but
sin^2 = 1 - cos^2
sin^2*sin^2 = 1-2cos^2+cos^4
(1-2cos^2+cos^4)^2
= 1-4cos^2x+6cos^4x-4cos^6x+cos^8x
recall that cos^2 x = (1+cos2x)/2
so, cos^4 x = (1+cos2x)^2/4 now cos^2 2x = (1+cos4x)/2
and so on. In the end, you get
sin^8(x)
= 1/128 (cos8x - 8cos6x + 28 cos4x - 56cos2x + 35)