What can I conclude from this inequality?

Question

I was solving an equation and reached this inequality $$t^2−1≤\sin^2\alpha$$

• What conclusions can I draw about $t$ from the above inequality (given that $\alpha \in \mathbb R$)?

• Is it correct to use the minimum value of RHS (zero) and proceed?

Here's the complete context of the question:

Answer 1

Since the minimum value of $\sin^2 \alpha$ is $0$, this means that for any real $t$ in $-1 ≤ t ≤ 1$, the inequality will hold regardless of the value of $\alpha$.

Similarly, the maximum value of $\sin^2 \alpha$ is $1$. This means that for any $t$ that satisfy $t < -\sqrt{2}$ or $t > \sqrt{2}$, the inequality will not hold regardless of $\alpha$.

Now you can relate this back to the range of the function.