# Trig Identities

## Homework Statement

1/1+cosx = csc^2x - cscxcotx

## The Attempt at a Solution

R.S= csc^2x - cscxcotx
= (1/sin^2x) - (1/sinx * cosx/sinx)
= (1/sin^2x) - (cosx/sinx^2x)
= 1-cosx / sin^2x
= 1 - cosx / 1 - cos^2x
= 1 / 1 - cos^2x

It does not match the left side and I am unsure of what I did incorrectly.

= 1 - cosx / 1 - cos^2x
= 1 / 1 - cos^2x
perhaps a simple typo error

see for yourself /check

perhaps a simple typo error

see for yourself /check
Where in the left side? Or in my actual work?

Where in the left side? Or in my actual work?

you have on the left side
1/(1+cosx)
and on right side you got ( 1-cosx) / (1-cos^2x);
the denominator can be written as (1+cosx).(1- cosx) using identity a^2- b^2 = (a+b).(a-b)
so the numerator gets cancelled and you get 1/(1+cosx)
am i right?

SteamKing
Staff Emeritus
Homework Helper

## Homework Statement

1/1+cosx = csc^2x - cscxcotx

## The Attempt at a Solution

R.S= csc^2x - cscxcotx
= (1/sin^2x) - (1/sinx * cosx/sinx)
= (1/sin^2x) - (cosx/sinx^2x)
= 1-cosx / sin^2x
= 1 - cosx / 1 - cos^2x
= 1 / 1 - cos^2x

It does not match the left side and I am unsure of what I did incorrectly.

This step here is the problem:

How did you go from

##=\frac{1-cos(x)}{1-cos^2(x)}## to

##=\frac{1}{1-cos^2(x)}## ?

drvrm
This step here is the problem:

How did you go from

##=\frac{1-cos(x)}{1-cos^2(x)}## to

##=\frac{1}{1-cos^2(x)}## ?
I got rid of one of the cos from numerator then got rid of one from denominator.

haruspex
Homework Helper
Gold Member
I got rid of one of the cos from numerator then got rid of one from denominator.
You certainly did not do that since the denominator did not change. But even if you had it would have been invalid.
In a fraction, you cannot add something to the numerator and balance that by adding the same thing to the denominator. All you can do is multiply (or divide) the numerator as a whole by something and do exactly the same to the denominator. This is what drvrm did in post #4. Even then, you do need to be a bit careful. If the thing you multiply or divide by might be zero under some circumstances then you should exclude that by writing something like "when ..... is not zero then...."

You certainly did not do that since the denominator did not change. But even if you had it would have been invalid.
In a fraction, you cannot add something to the numerator and balance that by adding the same thing to the denominator. All you can do is multiply (or divide) the numerator as a whole by something and do exactly the same to the denominator. This is what drvrm did in post #4. Even then, you do need to be a bit careful. If the thing you multiply or divide by might be zero under some circumstances then you should exclude that by writing something like "when ..... is not zero then...."
Okay thank you. When doing trig identities you are able to work on both sides correct?

haruspex