Trigonometric identities: sometimes, always, never

Question

My teacher today told us that sin²x-cos²x=1 is sometimes true. I thought it would be never true because it’s not equivalent to sin²x+cos²x=1. Could someone explain why this it is sometimes true?

Answer 1

Use x = 0

sin²(0) = 0

cos²(0) = 1

sin²x-cos²x = 0 - 1 = -1, it does not =1

But if you use x = pi/2, then

sin²x = 1 and cos²x = 0, this time

sin²x - cos²x = 1 - 0 =1, and so it does work.

This is why it is sometimes.