determine wheather each of the following is an identity or not prove it

1. cos^2a+sec^2a=2+sina
2. cot^2a+cosa=sin^2a

To determine whether each of the given equations is an identity or not, we need to simplify both sides of the equation and check if they are equal.

1. cos^2(a) + sec^2(a) = 2 + sin(a)

First, let's simplify the left side of the equation:
Using the Pythagorean identity, sec^2(a) = 1 + tan^2(a), we can rewrite the equation as:

cos^2(a) + 1 + tan^2(a) = 2 + sin(a)

Now, let's simplify further:
Using the identity tan^2(a) = sin^2(a) / cos^2(a), we can substitute it back into the equation:

cos^2(a) + 1 + sin^2(a) / cos^2(a) = 2 + sin(a)

Next, let's find the common denominator, which is cos^2(a):

cos^2(a) * cos^2(a) + cos^2(a) + sin^2(a) = 2cos^2(a) + sin(a) * cos^2(a)

Simplifying:

cos^4(a) + cos^2(a) + sin^2(a) = 2cos^2(a) + sin(a) * cos^2(a)

Using the Pythagorean identity, sin^2(a) = 1 - cos^2(a):

cos^4(a) + cos^2(a) + 1 - cos^2(a) = 2cos^2(a) + sin(a) * cos^2(a)

cos^4(a) + 1 = 2cos^2(a) + sin(a) * cos^2(a)

Now, we need to simplify both sides and see if they are equal.

cos^4(a) + 1 = cos^2(a) * (2 + sin(a))

Since the left and right sides of the equation are equal, we can conclude that the given equation is an identity.

2. cot^2(a) + cos(a) = sin^2(a)

First, let's simplify the left side of the equation:

Using the identity cot^2(a) = 1 / tan^2(a), we can rewrite the equation as:

1 / tan^2(a) + cos(a) = sin^2(a)

Again, using the identity tan^2(a) = sin^2(a) / cos^2(a), we substitute it back into the equation:

1 / (sin^2(a) / cos^2(a)) + cos(a) = sin^2(a)

Next, let's simplify further:

(cos^2(a) / sin^2(a)) + cos(a) = sin^2(a)

Now, we need to find the common denominator, which is sin^2(a):

cos^2(a) + cos(a) * sin^2(a) = sin^4(a)

Next, we simplify both sides of the equation and see if they are equal.

cos^2(a) + cos(a) * sin^2(a) = sin^4(a)

Since the left and right sides of the equation are not equal, we cannot conclude that the given equation is an identity.

Therefore, equation 1 is an identity, while equation 2 is not an identity.