Final answer:
The trigonometric identity (1+cosx)/(1-cosx) equals (cscx+cotx)^2 has been verified by simplifying the right-hand side using reciprocal and Pythagorean identities and showing it equals the left-hand side.
Step-by-step explanation:
To verify the identity (1+cosx)/(1-cosx) = (cscx+cotx)^2, let's start with the right-hand side of the equation and simplify it using trigonometric identities.
We know that cscx is the reciprocal of sinx, and cotx is the ratio of cosx to sinx. So, cscx + cotx can be written as 1/sinx + cosx/sinx, which simplifies to (1 + cosx)/sinx.
Now, let's square this expression: ((1 + cosx)/sinx)^2. This is the same as (1 + cosx)^2/sin^2x.
Using the Pythagorean identity sin^2x + cos^2x = 1, we can write sin^2x as 1 - cos^2x. Substituting into our equation, we get ((1 + cosx)^2)/(1 - cos^2x).
The denominator can be factored using the difference of squares formula, resulting in (1 + cosx)(1 - cosx). This simplifies the expression to (1 + cosx)/(1 - cosx), which is exactly the left-hand side of our original equation. Hence, we have verified the identity.