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.