Differentiating :calculus theory

[chapsticks]

Homework Statement

Verify the identity:

arctanx + arctan(1/x)=∏/2

using calculus theory.
(Hint: Differentiate the left hand side of the identity)

Homework Equations

?

The Attempt at a Solution

is this correct?

tan(arctanx + arctan(1/x))
= [tan(arctan(x)) + tan(arctan(1/x))][1 - tan(arctan(x))*tan(arctan(1/x))]

= [x + 1/x]/[1 - x*1/x] = (x + 1/x)/0 = oo

tan pi/2 = oo

 

[Quinzio]

mmm... they wanted you to do this:
\frac{1}{1+x^2}+ etc...

 

[chapsticks]

where did that come from I'm confused??

 

[Mark44]

Homework Statement

Verify the identity:

arctanx + arctan(1/x)=∏/2

using calculus theory.
(Hint: Differentiate the left hand side of the identity)

Homework Equations

?

The Attempt at a Solution

is this correct?

tan(arctanx + arctan(1/x))
= [tan(arctan(x)) + tan(arctan(1/x))][1 - tan(arctan(x))*tan(arctan(1/x))]

= [x + 1/x]/[1 - x*1/x] = (x + 1/x)/0 = oo

tan pi/2 = oo

Start by using the hint, which is to differentiate the left side.

 

[chapsticks]

f'(x)=1/(1+x²) + -1/(x²+1) =0

for all x≠0

function is constant on domain

is this right?

 

[Dick]

f'(x)=1/(1+x²) + -1/(x²+1) =0

for all x≠0

function is constant on domain

is this right?

Yes, it is. I guessing they meant to specify x>0. Now just pick a nice value of x to put into figure out what the constant is.