SOLUTION: Prove the identity: 1/(coseca-cota)-1/sina=1/sina-1/(coseca+cota)

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Question 1199469: Prove the identity:
1/(coseca-cota)-1/sina=1/sina-1/(coseca+cota)
Found 3 solutions by Edwin McCravy, MathTherapy, math_tutor2020:
Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
Prove the identity:



I'll simplify the left side:

1%2F%28csc%28a%29+-+cot%28a%29%29+-+1%2Fsin%28a%29



1%5E%22%22%2F%28%281-cos%28a%29%29%2Fsin%28a%29%29+-+1%5E%22%22%2Fsin%28a%29%5E%22%22

sin%28a%29%2F%281-cos%28a%29%29+-+1%2Fsin%28a%29

Multiply top and bottom of first fraction by its conjugate 1+cos(a)









%0D%0A%0D%0A%281%2Bcos%28a%29%29%0D%0A%0D%0A%2Fsin%28a%29+-+1%2Fsin%28a%29

%281%2Bcos%28a%29-1%29%2Fsin%28a%29

cos%28a%29%2Fsin%28a%29

cot%28a%29

Now you can finish.  Notice that the right side is just like the left 
side except for a few signs.  So simplify the right side exactly like 
I simplified the left side, and if you do it correctly, you'll get the 
same thing.

Edwin

Answer by MathTherapy(10556) About Me  (Show Source):
You can put this solution on YOUR website!
Prove the identity:

1/(coseca-cota)-1/sina=1/sina-1/(coseca+cota)

                                                  
 
   

                                                                        OR

                                                   
                 
                           highlight_green%281%2F%28csc+%28a%29+-+cot+%28a%29%29+-+1%2Fsin+%28a%29%29                                                highlight%281%2Fsin+%28a%29+-+1%2F%28csc+%28a%29+%2B+cot+%28a%29%29%29

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

The other tutors have provided great approaches.

I'll go over a visual way to confirm the identity.
This is meant to be a quick extra verification, and you should still do the algebra as mentioned by the other tutors.
The visual approach is merely a supplement.

In place of the variable 'a', I'll use x.

The given equation is

break up the equation into two functions

with f(x) and g(x) representing the left hand side and right hand side respectively.

Then use a graphing tool such as Desmos or GeoGebra.
Plotting f(x) and g(x) on the same xy grid will have the two curves line up perfectly.
One overlaps the other.
This visually confirms that f%28x%29 and g%28x%29 represent the exact same thing, and hence f%28x%29=g%28x%29 is an identity.

Click repeatedly to turn on and off one of the curves to have that curve blink, which shows the perfect overlap.
Use of different colors is strongly recommended.

Or we could plot h%28x%29+=+f%28x%29-g%28x%29
This will draw a completely flat horizontal line that is on top of the x axis.
Click to turn on and off the function curve, doing so repeatedly, so that you can see this overlap better.

Why is h%28x%29 on top of the x axis?
Well if f%28x%29+=+g%28x%29 for all x in the domain, then f%28x%29-g%28x%29+=+0 for all x in the domain.
It's like saying how 2x%2B3x+=+5x is the same as 2x%2B3x-5x+=+0 (which simplifies to 0+=+0)

Here's the link to the Desmos plot
https://www.desmos.com/calculator/olq3muwt7a
Ignore the blue and red vertical lines. They should NOT be there.
They are an unfortunate glitch since the calculator is trying to connect the dots to form a continuous curve; but each curved piece should be its own separate island.
A quick fix is to think of the vertical lines as asymptotes.
The h%28x%29 function in that Desmos graph is shown in green, perfectly aligned with the x axis.

Once again, you should use the methods mentioned by the other tutors.
This visual style is meant to be an extra verification step or supplement.
This visual method is informal; while the algebraic methods are formal and what your teacher is looking for.