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

Answer 1

A trig identity is an equation that is true for all values of x;

For example:

sin²x + cos²x will always equal 1 no matter what the value of x.

There are some equations considered to be identities that are in fact not true all the time.

tan(2x) = 2tanx/(1 - tan²x) is not true when x = π/2 (right side is undefined).

Check out some common trig identities here:

desmos.com/calculator/7zckcj0msy

A conditional equation is one that is true for a subset of the real numbers, i.e. for certain values of x, but not all.

For example:

sin²x - 1 = 0 is only true for some values of x.

sin²x = 1

sinx = ±√(1) = ±1

sinx = 1 or sinx = -1

x = sin^-11 or x = sin^-1(-1)

x = π/2 + 2kπ, k any integer or x = 3π/2 + 2kπ, k any integer (this can be condensed, but left this way for emphasis).