Question

How can I prove the following identity?

cos^(-1)x + cos^(-1) (-x) = pi

Answer 1

well
\[\cos^{-1} -x = \pi - \cos^{-1} x\]

Answer 2

Is that a trigonometric identity?

Answer 3

You could plug in the inverse,

\[\cos^{-1}(x) + \cos^{-1}(-x)\]
use \(x=\cos \theta\)

\[\cos^{-1}(\cos \theta) + \cos^{-1}(-\cos \theta)\]
The left hand side works out pretty easily like we wanted:


\[\theta + \cos^{-1}(-\cos \theta)\]

But what about that pesky right hand part? Well really if you look at the graph of cosine, negative cosine is just what happens when you shift the angle by pi,

\[-\cos(\theta) = \cos(-\theta + \pi)\]

If you look at the graphs and play around for a few seconds with changing the stuff inside like cos(x), cos(x+1), cos(x+2), etc... you'll get an intuition for some stuff. ANYWAYS plug that in and you get:

\[\theta -\theta + \pi\]

Which is what you wanted.

Answer 4

Ok, I think I understand. Thanks, I appreciate it.