Question

what questions do you have?

1. why would knowing the pythagorean identity $\sin^2 x + \cos^2 x = 1$ and the sum identities allow you to prove all the other identities you have seen in this lesson?
2. is $\sin(\sin^{-1} x) = \sin^{-1}(\sin x)$ an identity? why or why not?
3. how would you explain to someone who has not yet studied trigonometry the difference between an identity and an equation?
4. what questions do you still have about the unit? ask them here, and your classmates may give you the answers you seek.

Explanation:

Answer:

1. You can rearrange $\sin^2 x + \cos^2 x = 1$ (divide by $\cos^2 x$ or $\sin^2 x$) to derive $\tan^2 x + 1 = \sec^2 x$ and $1 + \cot^2 x = \csc^2 x$, and use sum identities to derive double-angle, half-angle, etc., identities.
2. No, it is not an identity. $\sin(\sin^{-1} x) = x$ for $x \in [-1, 1]$, while $\sin^{-1}(\sin x) = x$ only for $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$, so they are not equal for all valid inputs.
3. An identity is true for all values of the variable within its domain, while an equation is only true for specific values of the variable.
4. (Example question) How can we use the Pythagorean identity to simplify complex trigonometric expressions?