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Verify the following identities:
1/(1-sin x) + 1/(1+sin x) = 2sec^2 x
tan x + cot x = sec x csc x
sec y + tan y = cos y/(1 - sin y)
I'm having trouble with these last problems that I have to turn in before my test. I just can't seem to get the right answers.
Note: LHS = Left hand side, RHS = Right hand sideQuote:
Originally Posted by jenesaispas
1/(1-sin x) + 1/(1+sin x) = 2sec^2 x
Consider LHS
1/(1-sin x) + 1/(1+sin x)
= (1 + sinx + 1 - sinx)/(1 - sinx)(1 + sinx)
= 2/(1 - sin^2x)
= 2/cos^2x
= 2 sec^2x
=RHS
tan x + cot x = sec x csc x
Consider LHS
tan x + cot x
= sinx/cosx +cosx/sinx
= (sin^2x + cos^2x)/sinxcosx
= 1/sinxcosx
=(1/sinx)*(1/cosx)
= cscxsecx
= RHS
sec y + tan y = cos y/(1 - sin y)
Consider LHS
sec y + tan y
= 1/cosy + siny/cosy
= (cosy + cosysiny)/cos^2y
= (cosy + cosysiny)/(1 - sin^2y)
= [cosy(1 + siny)]/[(1 - siny)(1 + siny)].........the 1+siny cancels in the top and bottom
= cosy/(1 - siny)
= RHS
Now the thing with trig identities is that they require good visualization, inginuity, good algebra, and a touch of creativity--and they're loads of fun. but there are a few steps to make stupid people (like myself) get good at them too.Quote:
Originally Posted by jenesaispas
Tips:
You want to start working on the most complicated side, it gives you more options to work with.
You want to convert everything to sins and cosines, most people are familiar with them and so can manipulate them more skillfully, you can just change your answers to other trig functions when you're finished.
Not all trig identity problems work out as clean as these do. sometimes you have to work on one side and then work on the other side as well. If you can get both sides to the look the same by working them out a bit, you have proven the identity.
Have fun, dont get frustrated. there are usually several ways of doing each problem. one way can be a lot harder than another. if you get stuck, try a different manipulation, or try working on the other side. challenge yourself to come up with new ways of doing a single problem, see if you can get a solution in less lines than you did the first time, the practice will do you good, and you'll end up having more fun than you ever thought you could doing math.
If i did anything above that you dont understand, just say so