I am doing Parametric Equations in Calculus 2 where you want to convert a system of Parametric equations into Cartesian equations without the parameter t. By substituting $$\theta =\cos^{-1} x$$ in the Pythagorean Identity $$\sin^{2} \theta + \cos^{2} \theta=1$$ we get the relation: $$\sin{(\cos^{-1} x)}=\sqrt{1-x^{2}}$$ This particular identity is useful in eliminating $t$ from a system that represents a circle or an ellipse.
However, this gives me the thought that there are more identities like these (which I have seen earlier, but not paid attention to).
They don't seem to be covered in basic Trig books, but used where needed in Calculus courses.
Question 1: Can anyone give me a list of such identities (Internet references will work equally well). Having their proofs as well would be even better, since I am not always able to derive the proof myself.
Question 2: Where else are such identities useful? Even an example or an idea would be nice.
You can derive a trig function of an inverse trig function of x.
Suppose you want $\cos (\tan^{-1} x)$. Draw a right triangle, and label one acute angle $\theta$. Let $\theta = \tan^{-1} x$. So $\tan \theta = x = \frac{x}{1}$, and you can label the opposite $x$ and the adjacent $1$. Then by the Pythagorean theorem the hypotenuse is $\sqrt{1 + x^2}$, and so $\cos \theta = $ adjacent/hypotenuse = $\frac{1}{\sqrt{1+x^2}}$. This should give the correct formula for all $30$ different versions.