Rewrite the expression in terms of the first power of cosine
sin^4(x)
To rewrite the expression sin^4(x) in terms of the first power of cosine, we can use the identity:
sin^2(x) + cos^2(x) = 1
Let's represent sin^2(x) as 1 - cos^2(x):
sin^4(x) = (sin^2(x))^2
= (1 - cos^2(x))^2
Now, we can expand the square by using the formula (a - b)^2 = a^2 - 2ab + b^2:
(1 - cos^2(x))^2
= 1^2 - 2(1)(cos^2(x)) + (cos^2(x))^2
= 1 - 2cos^2(x) + cos^4(x)
So, sin^4(x) can be rewritten as:
sin^4(x) = 1 - 2cos^2(x) + cos^4(x)
To rewrite the expression sin^4(x) in terms of the first power of cosine, we can start by using the Pythagorean identity: sin^2(x) + cos^2(x) = 1. Rearranging this equation, we get sin^2(x) = 1 - cos^2(x).
Substituting this into the expression sin^4(x), we get:
(sin^2(x))^2 = (1 - cos^2(x))^2.
Expanding this expression, we have:
(1 - cos^2(x))^2 = 1^2 - 2(1)(cos^2(x)) + (cos^2(x))^2.
Simplifying further, we obtain:
1 - 2cos^2(x) + cos^4(x).
Therefore, sin^4(x) can be rewritten as 1 - 2cos^2(x) + cos^4(x) in terms of the first power of cosine.