trig idenity mishap

[markraz]

trig identity mishap

hello I need help with an intermediate step.
I have this trig function

cot(x)/(1+csc(x))

I'm trying to find the derivative
using quotient rule: note: ( (x) removed from trig functions for clarity)

f' g g' f
[(-csc^2)(1+csc)] - [(-csc cot) (cot)]
--------------------------------------
(1+csc)^2

I have up to here:
[-csc^2-csc^3] - [-csc cot^2]
--------------------------------------
(1+csc)^2


however the book shows this below: as the next step

-csc^2 -csc^3 + csc(csc^2 - 1)
--------------------------------------
(1+csc)^2



how do I get from green to blue? any hints?


final answer for reference:

-csc
-------
(1+csc)

thanks

 

[srmichael]

hello I need help with an intermediate step.
I have this trig function

cot(x)/(1+csc(x))

I'm trying to find the derivative
using quotient rule: note: ( (x) removed from trig functions for clarity)

f' g g' f
[(-csc^2)(1+csc)] - [(-csc cot) (cot)]
--------------------------------------
(1+csc)^2

I have up to here:
[-csc^2-csc^3] - [-csc cot^2]
--------------------------------------
(1+csc)^2


however the book shows this below: as the next step

-csc^2 -csc^3 + csc(csc^2 - 1)
--------------------------------------
(1+csc)^2



how do I get from green to blue? any hints?


final answer for reference:

-csc
-------
(1+csc)

thanks

You have to remember the trig identity: 1 + cot²x = csc²x

Carry on...

 

[Quaid]

ֺ
Hi Mark,

Another approach is to rewrite stuff in terms of sine and cosine.

Using your abbreviated notation:

-(-1/sin * cos^2/sin^2)

1/sin * cos^2/sin^2

1/sin * (1 - sin^2)/sin^2

1/sin * (1/sin^2 - sin^2/sin^2)

csc * (csc^2 - 1)


Also, when you can't seem to get from A to B, try going from B to A, instead. (You may always reverse the steps.)

Cheers

ֺ

 

[Quaid]

ֺ
PS: The answer you posted may be expressed as -1/[sin(x) + 1]

More practice, for you.

ֺ

 

[markraz]

ֺ
Hi Mark,

Another approach is to rewrite stuff in terms of sine and cosine.

Using your abbreviated notation:

-(-1/sin * cos^2/sin^2)

1/sin * cos^2/sin^2

1/sin * (1 - sin^2)/sin^2

1/sin * (1/sin^2 - sin^2/sin^2)

csc * (csc^2 - 1)


Also, when you can't seem to get from A to B, try going from B to A, instead. (You may always reverse the steps.)

Cheers

ֺ

excellent thanks

 

[markraz]

You have to remember the trig identity: 1 + cot²x = csc²x

Carry on...

thanks
in my problem how can this id quickly be identified?