Question

Prove that

Answer 1

\(cos^{-1}(-x)=\pi-cos^{-1}(x)\)

Answer 2

Let \(\alpha = \cos^{-1}(x) \implies \cos \alpha = x\)

\(\cos (\pi - \alpha ) = \cos \pi \cos \alpha + \sin \pi \sin \alpha = - \cos \alpha = - x\)

So:

\(\cos (\pi - \alpha ) = - x\)

\(\implies \pi - \alpha = \cos^{-1}(- x) \)

\(\implies \pi - \cos^{-1}( x) = \cos^{-1}(- x) \)

Or just draw it, easier that way