Answer :
The first step in simplifying is this one (since tangent is sin/cos, and sec and is the same as 1/cos):
[tex]-( \frac{sin ^{2}x }{cos ^{2} x} )+( \frac{1}{cos ^{2}x } )=1[/tex]
The next step is this:
[tex] \frac{1-sin ^{2} x}{cos ^{2}x } =1[/tex]
Now the next step in this identity involves knowing another identity, which is this one (a Pythagorean identity, actually);
[tex]sin ^{2} x+cos ^{2} x=1[/tex] and [tex]cos ^{2}x=1-sin ^{2} x[/tex]
So let's sub that in, like this:
[tex] \frac{cos ^{2}x }{cos ^{2}x } =1[/tex] which it does, right?
[tex]-( \frac{sin ^{2}x }{cos ^{2} x} )+( \frac{1}{cos ^{2}x } )=1[/tex]
The next step is this:
[tex] \frac{1-sin ^{2} x}{cos ^{2}x } =1[/tex]
Now the next step in this identity involves knowing another identity, which is this one (a Pythagorean identity, actually);
[tex]sin ^{2} x+cos ^{2} x=1[/tex] and [tex]cos ^{2}x=1-sin ^{2} x[/tex]
So let's sub that in, like this:
[tex] \frac{cos ^{2}x }{cos ^{2}x } =1[/tex] which it does, right?