How to explain to a friend the difference between a trigonometric identity and a conditional equation?

Answer 1

a trig identity is always true no matter what the angle is

cos^2x + sin^2x = 1 regardless of the value of x

sinx = 1 is only true for certain values of x. It is not a trig identity. It's true conditionally on what the value of x is.

you can solve for x: x = pi/2 + 2npi where n = any integer. sin(pi/2) = 1, sin(5pi/2) = 1