1) verify the identity

sec^2 (pi/2 - x) -1 = cot^2 x

I think that we can replace sec with csc for the cofunction formulas since there is pi/2. Since there is squared and a 1 I think we could use one of the pythagorean identites but I am not sure which one.

Did you notice that pi/2-x and x are complementary angles (add up to 90º)?

so we know that cos(pi/2-x) = sinx
I will use that ...

LS
= sec^2 (pi/2 - x) -1
= 1/cos^2 (pi/2-x) - 1
= 1/[cos(pi/2-x)cos(pi/2-x)] - 1
= 1/[sinxsinx] - 1
= 1/sin^2x - 1
= (1 - sin^2x)/sin^2x
= cos^2 x/sin^2 x
= cot^2 x
= RS

To verify the identity sec^2(pi/2 - x) - 1 = cot^2(x), we can start by expressing sec^2(pi/2 - x) and cot^2(x) in terms of sine and cosine using the cofunction identities.

First, let's replace sec with the cofunction identity, csc(pi/2 - x) = 1/sin(pi/2 - x). So our equation becomes:

(1/sin(pi/2 - x))^2 - 1 = cot^2(x)

Next, we need to find the cofunction identity for cot^2(x). Cotangent can be expressed as the reciprocal of tangent: cot(x) = 1/tan(x). Squaring both sides, we get:

cot^2(x) = (1/tan(x))^2

Now, let's rewrite tan(x) in terms of sine and cosine using the identity tan(x) = sin(x) / cos(x):

cot^2(x) = (1/(sin(x)/cos(x)))^2
= (cos(x)/sin(x))^2

Now we have expressed both sides of the equation in terms of sine and cosine. The next step is to simplify both sides using the Pythagorean identities.

The Pythagorean identity for sine is sin^2(x) + cos^2(x) = 1.

Let's rewrite the right-hand side of our equation cot^2(x) as (cos(x)/sin(x))^2:

cot^2(x) = (cos(x)/sin(x))^2
= cos^2(x)/sin^2(x)
= cos^2(x)/(1 - cos^2(x)) [using the Pythagorean identity sin^2(x) = 1 - cos^2(x)]

Now, let's simplify the left-hand side of our equation (1/sin(pi/2 - x))^2 - 1 using the Pythagorean identity.

The Pythagorean identity for cosine is cos^2(x) + sin^2(x) = 1.

Let's rewrite the left-hand side of our equation (1/sin(pi/2 - x))^2 - 1:

(1/sin(pi/2 - x))^2 - 1 = (1/(cos(x)))^2 - 1
= 1/cos^2(x) - 1

Now, we can see that both sides of the equation simplify to the same expression: cos^2(x)/(1 - cos^2(x)).

Therefore, we have verified the identity sec^2(pi/2 - x) - 1 = cot^2(x) by expressing both sides of the equation in terms of sine and cosine using cofunction identities, simplifying using Pythagorean identities, and showing that both sides simplify to the same expression.