How to solve for delta using trig functions? i.e. cosine sine tangent secant Thank you

I'm not sure what delta you're referring to. Could you please elaborate some for us. What is it you're trying to do with the trig funtions?

the problem is..

limit sec theda - 1/ theda sec theda
as theda approaches 0

find the limit

Ok, now I think I understand what you mean. We have:
"the problem is..

limit sec theda - 1/ theda sec theda
as theda approaches 0

find the limit"
This is the same as
limit theta->0 of (1/cos - 1)*cos/theta or,
limit theta->0 of (1-cos)/theta
You should have some limit theorems that justify the steps I did. It's just algebraic manipulation here.
Now evaluate the limit. You should have a result for this problem in your text. It's a very common one like sin/ theta.

x

To solve for the limit as theta approaches 0 of (1-cos(theta))/theta, we can make use of trigonometric identities and algebraic manipulation.

Step 1: Simplify the expression.
Using the identity cos(theta) = 1 - 2sin^2(theta/2), we can rewrite (1-cos(theta))/theta as (1 - (1 - 2sin^2(theta/2)))/theta.
This simplifies to (2sin^2(theta/2))/theta.

Step 2: Observe the form of the expression.
The expression (2sin^2(theta/2))/theta has a 0/0 form when theta approaches 0. This indeterminate form means that we can't directly substitute theta = 0 into the expression to find the limit.

Step 3: Apply L'Hopital's Rule.
L'Hopital's Rule allows us to differentiate the numerator and denominator separately in order to evaluate the limit.
Differentiating the numerator, we get 4sin(theta/2)cos(theta/2).
Differentiating the denominator, we get 1.

Step 4: Evaluate the new expression.
Applying L'Hopital's Rule again, (4sin(theta/2)cos(theta/2))/1 simplifies to 2sin(theta/2)cos(theta/2).

Step 5: Evaluate the limit.
Now, we can substitute theta = 0 into the expression.
As theta approaches 0, sin(theta/2) approaches 0 and cos(theta/2) approaches 1.
Therefore, the limit of (2sin(theta/2)cos(theta/2))/1 as theta approaches 0 is equal to 0.

Therefore, the limit of (1-cos(theta))/theta as theta approaches 0 is 0.