Question

Why is `cosx=secx` an inconsistent equation?

Answer 1

Proof by counter-example, substituting a value for x.

Explanation:

`secx=1/cosx`

This means that we can write the above statement as:

`cosx=1/cosx`

This is quite easy to disprove.
Let `x=30^o`

`cos30^o=sqrt3/2`
`1/(cos30^o) =2/sqrt3`
(I can't be bothered to rationalize and I don't need to).

`2/sqrt3 != sqrt3/2 :. cos30^o!=1/(cos30^o), AAx inR`

However, some values of `x` do work.

By rearranging:
`cos^2x=1`
`cosx=+-sqrt(1)=+-1`

`x=arccos(1)=[(0^circ,360^circ,720^circ,cdots,n360^circ),(0, 2pi,4pi,cdots,2npi)]`, or
`x=arccos(-1)=[(180^circ,540^circ,900^circ,cdots,180^circ+n360^circ),(pi,3pi,5pi,cdots,pi+2npi)]`