State whether or not the equation is an identity. Prove your statement.

sin(x)/cos(x)+ cos(x)/1+sin(x)=sec(x)
(sorry about that)

To determine if the equation is an identity, we need to prove that both sides of the equation are equal for all possible values of x.

Let's start by simplifying both sides of the equation.

On the left side:
Using the common denominator cos(x)(1 + sin(x)), we simplify the expression:
sin(x)/cos(x) + cos(x)/(1 + sin(x)) = [(sin(x)(1 + sin(x)) + cos^2(x))/(cos(x)(1 + sin(x)))]

On the right side:
The right side is given as sec(x), which is equal to 1/cos(x).

Now, let's simplify the right side to have a common denominator with the left side:
1/cos(x) = [(1/cos(x))(cos(x)(1 + sin(x)))] / [(cos(x)(1 + sin(x)))]
= [(cos(x)(1 + sin(x)))/(cos(x)(1 + sin(x)))]
= 1

Now, we compare the simplified forms of both sides of the equation:

(simplified left side) [(sin(x)(1 + sin(x)) + cos^2(x))/(cos(x)(1 + sin(x)))]
= [(sin(x) + sin(x)^2 + cos^2(x))/(cos(x)(1 + sin(x)))]

The trigonometric identity sin^2(x) + cos^2(x) = 1 can be used to simplify the numerator:
= [(sin(x) + 1)/(cos(x)(1 + sin(x)))]

Now, we compare this simplified left side with the simplified right side, which is equal to 1:

[(sin(x) + 1)/(cos(x)(1 + sin(x)))] = 1

Since both sides of the equation are equal, for all possible values of x, we can conclude that the initial equation is an identity.