show that:
2^5 cos^2a sin^4a = cos6a-2cos4a- cos2a + 2
To show that 2^5 cos^2(a) sin^4(a) is equal to cos(6a) - 2cos(4a) - cos(2a) + 2, we can use trigonometric identities and algebraic manipulation.
First, let's use the double-angle identity for cosine and express cos^2(a) and sin^2(a) in terms of cos(2a):
cos(2a) = 2cos^2(a) - 1
cos^2(a) = (1 + cos(2a)) / 2
Next, let's substitute this value into the expression:
2^5 cos^2(a) sin^4(a) =
2^5 * [(1 + cos(2a)) / 2] * sin^4(a)
Since sin^2(a) = 1 - cos^2(a), we can express sin^4(a) as:
sin^4(a) = (1 - cos^2(a))^2
Substitute sin^4(a) with its equivalent expression:
2^5 * [(1 + cos(2a)) / 2] * [(1 - cos^2(a))^2]
Now let's simplify this expression:
2^5 / 2 * (1 + cos(2a)) * (1 - cos^2(a))^2 =
2^4 * (1 + cos(2a)) * (1 - cos^2(a))^2
Let's rewrite cos(2a) using the double-angle identity for cosine:
cos(2a) = 2cos^2(a) - 1
The expression becomes:
2^4 * (1 + (2cos^2(a) - 1)) * (1 - cos^2(a))^2 =
2^4 * (2cos^2(a)) * (1 - cos^2(a))^2
Now, let's multiply the terms:
2^4 * 2 * (cos^2(a)) * (1 - cos^2(a))^2 =
2^5 * (cos^2(a)) * (1 - cos^2(a))^2
Using the difference of squares, (1 - cos^2(a))^2 can be simplified as:
(1 - cos^2(a))^2 = sin^4(a)
Substituting this value back into the expression:
2^5 * (cos^2(a)) * (sin^4(a)) =
2^5 * (cos^2(a)) * sin^4(a)
Now, let's express cos^2(a) as 1 - sin^2(a):
2^5 * (1 - sin^2(a)) * sin^4(a) =
2^5 * (sin^6(a) - sin^2(a) * sin^4(a)) =
2^5 * (sin^6(a) - sin^6(a) * sin^2(a)) =
2^5 * sin^6(a) * (1 - sin^2(a))
Using another trigonometric identity, sin^2(a) = 1 - cos^2(a), we can rewrite (1 - sin^2(a)) as cos^2(a):
2^5 * sin^6(a) * cos^2(a) =
2^5 * sin^6(a) * (1 - cos^2(a) + cos^2(a)) =
2^5 * sin^6(a) * (sin^2(a) + cos^2(a)) =
2^5 * sin^6(a)
We know that sin^6(a) = sin^2(a) * sin^4(a) = sin^4(a) * (1 - cos^2(a)).
Substituting this back into the expression:
2^5 * sin^6(a) =
2^5 * sin^4(a) * (1 - cos^2(a)) =
2^5 * (sin^2(a) * (1 - cos^2(a))) * (1 - cos^2(a)) =
2^5 * sin^2(a) * (1 - cos^2(a))^2
Notice that (1 - cos^2(a))^2 is equal to sin^4(a), so the final expression becomes:
2^5 * sin^2(a) * sin^4(a) =
2^5 * sin^6(a)
Therefore, we have proven that 2^5 * cos^2(a) * sin^4(a) is equal to 2^5 * sin^6(a).
It's worth noting that cos(6a) - 2cos(4a) - cos(2a) + 2 is not equal to 2^5 * sin^6(a). The expression you provided is not equivalent to the original expression.