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Trigonometric identity example proof involving sec, sin, and cos

Let's try to prove a trigonometric identity involving Secant, sine, and cosine of an angle to understand how to think about proofs in trigonometry. There are many ways to prove these identities. Created by Aanand Srinivas.

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• How would you solve "1- [cos x squared/1 +sin x]?
• I think this is a late answer, but anyway, here you are:
1- [cosX squared/1 +sinX]
= [1 + sinX - (cosX)^2]/1 + sinX

This is possible because 1 = (1 + sinX)/(1 + sinX)
We also know that 1 - (cosX)^2 = (sinX)^2
Now,

[1 - (cosX)^2 + sinX]/1 + sinX
= [(sinX)^2 + sinX]/1 + sinX

Factoring out the sinX,

[sinX (sinX + 1)]/1 + sinX
= [sinX (1 + sinX)]/1 + sinX
= sinX

Hope this helped!
• I got lost for a brief moment, how did you turn that 1 on the right side into 1^2. Do I have to do anything to the equation to achieve that? Or can I write as 1^2 just because I want to?
• Good question!
No, you don't have to do anything to the equation to achieve that, since 1 is equal to 1 squared.
The only reason it is written like that in the video is to portray the a^2 - b^2 format, which can be simplified into (a + b)(a - b), as was done.
Hope this helped!