
Doug C. answered 12/29/17
Math Tutor with Reputation to make difficult concepts understandable
Hi Gnarls,
Since tan = sin/cos and sec = 1/cos you can write:
(sinx/cosx) + (1/cosx)
--------------------------
cosx
Think of this as a complex fraction which can be simplified by multiplying top and bottom of the "big" fraction by cos x.
That results in :
(sinx + 1)
------------
(cos2x)
Replace the denominator by 1 - sin2x and change the order of the terms in the numerator.
(1 + sin x)
--------------
1 - sin2x
Finally the denominator is a difference of squares so can be written as (1+sinx)(1-sinx). This results in the (1+sinx) factor in numerator and denominator cancelling out.
The result: 1/(1-sinx)
Here is some proof:
https://www.desmos.com/calculator/3qtyexkalx
The graphs of both f and g are identical.