Question

How do you prove `secx - cosx = tanx * sinx`?

Answer 1

Using the definitions `sec(x)=1/cos(x)` and `tan(x)=sin(x)/cos(x)` along with the identity `sin^2(x)+cos^2(x)=1 => sin^2(x)=1-cos^2(x)`,
for `cos(x)!=0` we have

`sec(x)-cos(x) = 1/cos(x)-cos^2(x)/cos(x)`

`=(1-cos^2(x))/cos(x)`

`=sin^2(x)/cos(x)`

`=sin(x)/cos(x)*sin(x)`

`=tan(x)*sin(x)`