frac(cos(theta))(1+sin(theta)) + frac(1+sin(theta))(cos(theta))
Multiply the left fraction by cos(theta) and the right fraction by (1+sin(theta)) to put everything under the same denominator:
= frac((cos(theta))^2 +(1+sin(theta))^2)((1+sin(theta))*cos(theta))
Then we open the parentheses (at the numerator, just leave the denominator untouched) and get:
= frac(cos^2(theta)+1+2sin(theta)+sin^2(theta))((1+sin(theta))*cos(theta))
You should remember the identity cos^2(theta)+sin^2(theta) =1 which you can see in the numerator. So we get:
= frac(1+2sin(theta)+1)((1+sin(theta))*cos(theta))
which simplifies to:
= frac(2+2sin(theta))((1+sin(theta))*cos(theta))
but we can factor out the 2 so:
= 2*frac(1+sin(theta))((1+sin(theta))*cos(theta))
Now remove the (1+sin(theta)) from top (numerator) and bottom (denominator) (you can do this because any number over itself is 1).
Then you are left with:
= 2*frac(1)(cos(theta))
and since frac(1)(cos(theta))=sec(theta) by definition of the secant you do indeed have:
= 2*sec(theta)
Q.E.D.
