How can the following two identities be verified?
1. (cos^2y)/(1-siny)=1+siny
2. sin^2a-sin^4a=cos^2a-cos^4a
Thanks!
change cos^2a to 1-sin^2a=(1-sina)(1+sina)
Thank you. I figured out the first one with your help, but I'm still stuck on the second. Any tips?
To verify the second identity, we will use the Pythagorean identity (sin^2x + cos^2x = 1) and some algebraic manipulations.
Starting with the left-hand side (LHS) of the second identity:
sin^2a - sin^4a
We can rewrite sin^2a as (1 - cos^2a) using the Pythagorean identity:
(1 - cos^2a) - sin^4a
Next, let's move the terms with cos^2a together:
1 - cos^2a - sin^4a
To simplify this, we can rewrite sin^4a as (sin^2a)^2:
1 - cos^2a - (sin^2a)^2
Now, we can apply the identity you mentioned earlier:
1 - cos^2a - (1 - cos^2a)^2
Expanding the squared term:
1 - cos^2a - (1 - 2cos^2a + cos^4a)
Combining like terms:
1 - cos^2a - 1 + 2cos^2a - cos^4a
Simplifying further:
cos^2a - cos^4a
Now, we can compare this with the right-hand side (RHS) of the second identity:
cos^2a - cos^4a
Since the LHS and RHS of the identity are equal, we have verified the second identity.
So, the second identity can be verified by using the Pythagorean identity and algebraic manipulations to simplify and show that the LHS is equal to the RHS.