How do you prove Sec(x) - cos(x) = sin(x) * tan(x)?

1 Answer
Jul 10, 2016

Knowing that sec(x) = 1/cos(x) and tan(x) = sin(x)/cos(x), rewriting the equation yields

1/cos(x) - cos(x)/1 = sin(x) * (sin(x)/cos(x))

Rewriting the left fraction by using the property

a/b-c/d =(ad-bc)/(bd)

and simplifying the right side of the equation yields

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

Note that in this case, we can make use of the identity

sin^2(x)+cos^2(x)=1, since 1-cos^2(x)=sin^2(x), giving us

sin^2(x)/cos(x)=sin^2(x)/cos(x), which is true, therefore we have proven that

sec(x)-cos(x) = sin(x) tan(x)