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Calculus of inverse trig

Hi, I've got this question that's got me stumped. How do I show, given

y=\arctan (\frac{1+x}{1-x})

that

\frac {dy}{dx}=\frac{1}{1+x^2}

,
and hence show that

\arctan (\frac{1+x}{1-x})-arctanx=\pi/4

for x<1
Thanks

Reply 1

EconFan_73

tany = (1+x)/(1-x)

I think that's the step you're probably struggling to recognise :smile:

Reply 2

foorganders

Original post by EconFan_73

tany = (1+x)/(1-x)

I think that's the step you're probably struggling to recognise :smile:

I've got that, but I wasn't sure what to do from there. I recall integrating both sides at this step but that doesnt seem to lead me anywhere

Reply 3

ghostwalker

Original post by foorganders

I've got that, but I wasn't sure what to do from there. I recall integrating both sides at this step but that doesnt seem to lead me anywhere

Try differentiating the LHS of what you're trying to show. What does it equal? Hence the original expression is....

Reply 4

foorganders

Original post by ghostwalker

Try differentiating the LHS of what you're trying to show. What does it equal? Hence the original expression is....

I'm not sure which part you mean

Reply 5

ghostwalker

Original post by foorganders

I'm not sure which part you mean

The

\arctan (\frac{1+x}{1-x})-arctanx

Edit: LHS stands for "left hand side" - fairly standard and endemic in maths.

(edited 9 years ago)

Reply 6

foorganders

Original post by ghostwalker

The

\arctan (\frac{1+x}{1-x})-arctanx

Edit: LHS stands for "left hand side" - fairly standard and endemic in maths.

I know that... I thought you meant something in the first part, which is the bit I'm focused on for now

Reply 7

ghostwalker

Original post by foorganders

I know that... I thought you meant something in the first part, which is the bit I'm focused on for now

OK. If you're stuck on the tany=..., then you need to differentiate using implicit differentiation.

PS: It would help to simplify the RHS by polynomial division first.

(edited 9 years ago)

Reply 8

foorganders

Original post by ghostwalker

OK. If you're stuck on the tany=..., then you need to differentiate using implicit differentiation.

PS: It would help to simplify the RHS by polynomial division first.