Question

verify the identity
cosx/1-cscx=cscx-1/cotx

Answer 1

I know that I have to add sine to the equation... but thats about it

Answer 2

No. To prove an identity, you cannot work across the equal sign

Answer 3

\[\frac{\cos x}{1-cosec\ x}\]
Multiply numerator and denominator by (1+cosec x )
\[\frac{\cos x}{1-cose\ x}\times \frac{1+cosec\ x}{1+cosec\ x}\]
\[\frac{\cos x \times (1+cosec\ x)}{1-cosec^2 x}\]
We know that
\[1-cosec^2 x=-cot^2 x\]
\[\frac{\cos x \times (1+cosec\ x)}{-\cot^2 x}\]
Can you try from here?

Answer 4

okay let me try from here
but what about the other side?...like I don't exactly get what I am trying to accomplish?

Answer 5

Only work on one side, I'm sure you'll get the same expression as right side

Answer 6

okay wait.. does cos become 1/secx?

Answer 7

cosx I mean

Answer 8

@ash2326

Answer 9

Yes

Answer 10

okay so then I will have
\[1/secx*(1+cosecx) / (-\cot^2x)\]

Answer 11

@ash2326

Answer 12

I honestly have no clue what to do after...

Answer 13

\[\frac{\cos x \times (1+cosec\ x)}{-\cot^2 x}\]
\[\frac{\cos x \times (1+cosec\ x)}{-\cot x\times \frac{\cos x}{\sin x}}\]
Now we get
\[\frac{\sin x \times (1+cosec\ x)}{-\cot x}\]
\[\frac{\sin x+ 1}{-\cot x}\]
Do you understand till here?

Answer 14

the only thing I don't get is where did cosec x go?

Answer 15

sin x =1/cosec x

Answer 16

oh so it cancels out?

Answer 17

yes

Answer 18

okay so I get sinx+1/-cotx = -cotx/sinx+1?

Answer 19

@ash2326

Answer 20

I could see that you have put the identity wrong, if you put x=45 degrees. You can verify that left side is not equal to right side
\[\frac{\frac{1}{\sqrt 2}}{1-\sqrt 2}\ne\frac{\sqrt 2-1}{1}\]
Are you sure that the question is correct?

Answer 21

verify the identity cosx/1+cscx=cscx-1/cotx
its positive on the left side.. >.< sorry

Answer 22

So we have to multiply the left side by (1-cosec x )
Now proceed the way I did.