(i)prove that 1/1+sintheta +1/1-sin theta=2sec^2 theta
I'll just use x for ease of typing
(note the use of parentheses to make things clear!)
1/(1+sinx) + 1/(1-sinx)
=((1-sinx)+(1+sinx))/((1+sinx)(1-sinx))
= 2/(1-sin^2(x))
= 2/cos^2(x)
= 2sec^2(x)
By far the most useful trig identity is
sin^2(x) + cos^2(x) = 1
It pops up in many many disguises.