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

1 Answer
Jorge G. Pires ยท Daniel L.
Apr 23, 2015

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

Explanation:

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

Related questions
See all questions in Proving Identities
Impact of this question
60685 views around the world
You can reuse this answer
Creative Commons License