How do you prove ## [(1)/(1-sinx)]+[(1)/(1+sinx)]=2sec^2x##?

The formula can be proven by applying: 1) Least common multiple; 2) applying the trigonometric entity ##sin^2x + cos^2x=1 ##

Head

Key-relation : ##sin^2x + cos^2x=1##

Key-concept: Least common multiple; when no common multiples, just multiply the terms in the denominator.

Calculation

The above formula can be proven by transforming left side to right side:

##1/(1-sin x)+1/(1+sin x)= (1+sin x + 1-sin x)/((1+sinx)(1-sinx))##

To arrive to right-hand side, just divide the denominator to ##(1+sinx)(1-sinx) ##, the least common multiple, and multiply the numerator to the remaining, since they are all 1, just put the value.

By simple algebra and make use of ##(a-b)(a+b)=a^2 - b^2 ##, it can be seen from normal multiplication.

## (1+sin x + 1-sin x)/((1+sinx)(1-sinx))= 2/(1-sin^2x)##

Finally apply: ##sin^2x + cos^2x=1##, which gives out ##cos^2x=1 - sin^2x ##

## 2/(1-sin^2x)=2/cos^2x=2*(1/cosx)^2##

To finish, remember that ## secx=1/cosx##, hence:

## 2*(1/cosx)^2=2sec^2x##