Given ^-1(x) = alpha and ^-1(x) = beta, show that (alpha + beta) = 1.

Let's knock this out.

Question 4:

• Part (i): Given $\sin^{-1}(x) = \alpha$ and $\cos^{-1}(x) = \beta$, show that $\sin(\alpha + \beta) = 1$.

  Step 1: From the given information, we have:

  $$x = \sin(\alpha)$$ $$x = \cos(\beta)$$

  Step 2: Recall the identity for inverse trigonometric functions:

  $$\sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}$$

  Step 3: Substitute $\alpha$ and $\beta$ into the identity:

  $$$$