## Need help verifying an identity - sums and differences

Trigonometric ratios and functions, the unit circle, inverse trig functions, identities, trig graphs, etc.
staycw71
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Joined: Wed Nov 23, 2016 11:09 pm
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### Need help verifying an identity - sums and differences

Can anyone help me verify this identity?

tan (a-b) = (1 - cot(a) tan(b)) / cot(a) + cot(b)

Thank you

staycw71
Posts: 2
Joined: Wed Nov 23, 2016 11:09 pm
Contact:

### Need help verifying an identity - sums and differences

I apologize if this is a duplicate post. I thought I posted it already, but I cannot find it anywhere.

Can anyone help me verify the identity below?

tan (a-b) = (1 - cot(a )tan(b)) / cot(a) + tan(b)

little_dragon
Posts: 226
Joined: Mon Dec 08, 2008 5:18 pm
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### Re: Need help verifying an identity - sums and differences

Can anyone help me verify this identity?

tan (a-b) = (1 - cot(a) tan(b)) / cot(a) + cot(b)
they show how to do this stuff here: http://www.purplemath.com/modules/proving.htm
the right hand side (RHS) is messier so try starting there
see if maybe the fraction simplifies

1 - cot(a)tan(b) = 1 - cot(a)/cot(b) = [cot(b) - cot(a)] / cot(b)

then the whole thing is

[cot(b) - cot(a)] / cot(b)
----------------------------
cot(a) + cot(b)

cot(b) - cot(a)
---------------------------
cot(b) [cot(a) + cot(b)]

maybe this isn't going anywhere?
what happens if we turn this into tangents?

1/tan(b) - 1/tan(a)
-----------------------------
[1/tan(b)] [1/tan(a) + 1/tan(b)]

[tan(a) - tan(b)] / [tan(b)tan(a)]
---------------------------------------
[1/tan(b)] [{tan(b) + tan(a)} / tan(b)tan(a)]

move stuff around

[tan(a) - tan(b)} / [tan(a)]
---------------------------------------
[{tan(b) + tan(a)} / tan(b)tan(a)]

tan(a) - tan(b)
------------------------------
[tan(b) + tan(a)] / tan(b)

tan(a) - tan(b)
--------------------
1 + tan(a)/tan(b)

this is close to the right identity but not quite
is that one tangent in the original maybe supposed to be a cotangent?
thanks