Let’s solve the given problems:

35. $\cos(\sin^{-1}(x))$

We know that if $\sin^{-1}(x) = \theta$, then $\sin(\theta) = x$. Using the Pythagorean identity:

$\cos^2(\theta) + \sin^2(\theta) = 1$

$\cos^2(\theta) = 1 - \sin^2(\theta)$

$\cos(\theta) = \sqrt{1 - x^2}$

Thus:

$\cos(\sin^{-1}(x)) = \sqrt{1 - x^2}$

36. $\sin(\tan^{-1}(x))$

If $\tan^{-1}(x) = \theta$, then $\tan(\theta) = x$. From the right triangle definition, the sides are:

• Opposite = $x$
• Adjacent = $1$
• Hypotenuse = $\sqrt{x^2 + 1}$

$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{x}{\sqrt{x^2 + 1}}$

Thus:

$\sin(\tan^{-1}(x)) = \frac{x}{\sqrt{x^2 + 1}}$

37. $\tan(\sin^{-1}(x))$

If $\sin^{-1}(x) = \theta$, then $\sin(\theta) = x$. From the Pythagorean identity:

• Opposite = $x$
• Hypotenuse = $1$
• Adjacent = $\sqrt{1 - x^2}$

$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{x}{\sqrt{1 - x^2}}$

Thus:

$\tan(\sin^{-1}(x)) = \frac{x}{\sqrt{1 - x^2}}$

38. $\cos(\tan^{-1}(x))$

If $\tan^{-1}(x) = \theta$, then $\tan(\theta) = x$. From the triangle:

• Opposite = $x$
• Adjacent = $1$
• Hypotenuse = $\sqrt{x^2 + 1}$

$\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{1}{\sqrt{x^2 + 1}}$

Thus:

$\cos(\tan^{-1}(x)) = \frac{1}{\sqrt{x^2 + 1}}$

Summary:

1. $\cos(\sin^{-1}(x)) = \sqrt{1 - x^2}$
2. $\sin(\tan^{-1}(x)) = \frac{x}{\sqrt{x^2 + 1}}$
3. $\tan(\sin^{-1}(x)) = \frac{x}{\sqrt{1 - x^2}}$
4. $\cos(\tan^{-1}(x)) = \frac{1}{\sqrt{x^2 + 1}}$

Let me know if you'd like further explanation!

Relative Questions:

1. How do trigonometric identities relate to inverse functions?
2. What is the geometric interpretation of the inverse trigonometric functions?
3. How can these results be used in applied problems involving angles?
4. What happens if $x$ exceeds the domain of certain inverse trigonometric functions?
5. How can we generalize these results to different coordinate systems?

Tip: Always draw a right triangle when dealing with inverse trigonometric expressions to clearly visualize the relationships between sides and angles.