1)Use the definitions sin0(theta) = y/r, cos0(theta) = x/r and/or tan0(theta) = y/x to prove that

cot0(theta) = 1/tan0(theta)

2)For all 0(theta) prove:
a)cos(t) = 1/sec0(theta)
b. sec(t) = 1/cos0(theta)

1) To prove that cot0(theta) = 1/tan0(theta), we can start by using the definitions of tan0(theta) and cot0(theta).

From the definition of tan0(theta), we have:
tan0(theta) = y/x

From the definition of cot0(theta), we have:
cot0(theta) = x/y

To prove that cot0(theta) = 1/tan0(theta), we can rewrite cot0(theta) as:
cot0(theta) = x/y
= 1/(y/x)
= 1/tan0(theta)

Therefore, cot0(theta) = 1/tan0(theta).

2) For all 0(theta) prove:
a) To prove that cos(t) = 1/sec0(theta), we can use the definitions of cos0(theta) and sec0(theta).

From the definition of cos0(theta), we have:
cos0(theta) = x/r

From the definition of sec0(theta), we have:
sec0(theta) = 1/cos0(theta)

To prove that cos(t) = 1/sec0(theta), we can rewrite 1/sec0(theta) as:
1/sec0(theta) = 1/(1/cos0(theta))
= cos0(theta)

Therefore, cos(t) = 1/sec0(theta).

b) To prove that sec(t) = 1/cos0(theta), we can again use the definitions of cos0(theta) and sec0(theta).

From the definition of cos0(theta), we have:
cos0(theta) = x/r

From the definition of sec0(theta), we have:
sec0(theta) = 1/cos0(theta)

To prove that sec(t) = 1/cos0(theta), we can rewrite sec(t) as:
sec(t) = 1/(x/r)
= r/x
= 1/cos0(theta)

Therefore, sec(t) = 1/cos0(theta).

To prove the given identities, we will use the definitions sin(theta) = y/r, cos(theta) = x/r, and tan(theta) = y/x.

1) Proof of cot(theta) = 1/tan(theta):

First, let's express cot(theta) in terms of sin(theta) and cos(theta):

cot(theta) = cos(theta) / sin(theta)

Now substitute cos(theta) = x/r and sin(theta) = y/r:

cot(theta) = (x/r) / (y/r)

Dividing by a fraction is the same as multiplying by its reciprocal, so we can rewrite the expression as:

cot(theta) = (x/r) * (r/y)

The 'r' factors cancel out:

cot(theta) = x/y

Finally, we know that tan(theta) = y/x, so we can substitute tan(theta) into our expression:

cot(theta) = 1 / tan(theta)

Therefore, we have proved that cot(theta) = 1/tan(theta).

2) Proofs for the given identities:

a) Proof of cos(t) = 1/sec(theta):

We know that sec(theta) = 1/cos(theta). Multiplying both sides by cos(theta), we get:

cos(theta) * sec(theta) = 1

Now divide both sides by cos(theta):

cos(theta) * (1/cos(theta)) = 1/cos(theta)

The cos(theta) factors cancel out:

1 = 1/cos(theta)

Therefore, we have proved that cos(theta) = 1/sec(theta).

b) Proof of sec(theta) = 1/cos(theta):

We know that cos(theta) = x/r. Taking the reciprocal of both sides, we get:

1/cos(theta) = r/x

But we also know that sec(theta) = r/x, so we have:

sec(theta) = 1/cos(theta)

Thus, we have proved that sec(theta) = 1/cos(theta).

Note: In these proofs, we used the basic definitions of trigonometric functions and algebraic manipulations to derive the desired identities. It is important to remember the definitions and algebraic properties to prove trigonometric identities accurately.