Have you ever wondered what is expressed just in x? Probably not, but it might actually sometimes be useful to know. The expression might for example turn up when you’ve done a trigonometric substitution in an integral. So I decided how to go about to show the trigonometric identities
The idea is the use the well known identity and try to get a relation between the basic trig functions and the arcus functions trough that. It goes like this
using the well known identity. We see that we already have the wanted relation, so that the rest is algebra. By squaring both sides we get
And there we have the first one. Similarly to obtain the next one we go
Obtaining the symmetric identity between sine and cosine. For tangent we go
where we separately use expression 3 and 4 in the equal-chain above to obtain
and then for nr.4
And that was that. Later i will do etc.. but that requires complex numbers.