prove the identity: cos^4 - sin^4 = cos^2 - sin^2
(cos^2 + sin^2)(cos^2 - sin^2)
cos^2 + sin^2 = 1
cos^2 - sin^2 = cos^2 - sin^2
is this correct?
No, the steps you provided do not correctly prove the given identity. Here is the correct step-by-step explanation:
Starting with the left-hand side (LHS) of the given equation:
LHS: cos^4 - sin^4
We can rewrite cos^4 as (cos^2)^2 and sin^4 as (sin^2)^2:
LHS: (cos^2)^2 - (sin^2)^2
Now, we can use the identity x^2 - y^2 = (x + y)(x - y) to factorize:
LHS: [(cos^2 + sin^2)(cos^2 - sin^2)]
Since cos^2 + sin^2 is equal to 1 (due to the Pythagorean identity), we have:
LHS: [1(cos^2 - sin^2)]
Finally, we can rewrite 1(cos^2 - sin^2) as just (cos^2 - sin^2):
LHS: cos^2 - sin^2
Now, comparing the LHS (cos^2 - sin^2) with the right-hand side (RHS) (cos^2 - sin^2), we can conclude that they are equal.
Therefore, the identity is proved: cos^4 - sin^4 = cos^2 - sin^2.
Yes, your steps are correct. Let's go through them in more detail:
We start with the expression cos^4(x) - sin^4(x).
To prove the identity, we use the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b).
In this case, a stands for cos^2(x) and b stands for sin^2(x).
So, we rewrite cos^4(x) - sin^4(x) as (cos^2(x) + sin^2(x))(cos^2(x) - sin^2(x)).
Now, we simplify the first term, cos^2(x) + sin^2(x), which is equal to 1 by the Pythagorean identity: sin^2(x) + cos^2(x) = 1.
Substituting this simplification back into our expression, we get (1)(cos^2(x) - sin^2(x)) = cos^2(x) - sin^2(x).
Therefore, your simplification cos^2(x) - sin^2(x) is correct, and you have proven the identity.