Question 836438
Do you know what tan and arc tan mean ?
If not, {{{ tan(a) }}} is the side opposite angle {{{ a }}}
divided by the side adjacent to angle {{{ a }}}.
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Of course, You are always talking about a right triangle
{{{ arc tan(b) }}} means " the angle whose tan is {{{ b }}}.
{{{ b }}} is a number here, not an angle
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{{{ tan( arc tan( 1/2 ) + arc tan( 1/3 ) ) }}}
Work inside the parentheses one term at a time.
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{{{ arc tan( 1/2 ) }}}
On my Casio fx-115 ES, I press [ SHIFT-tan- 1/2 = ]
That gives me {{{ 26.565 }}} which is the angle whose
tan is {{{ 1/2 }}}
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Next I press [ SHIFT-tan-1/3 = ] 
That give me {{{ 18.435 }}} which is the angle whose
tan is {{{ 1/3 }}}
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Now add the angles:
{{{ 26.565 + 18.435 = 45 }}}
At this point, you have:
{{{ tan( arc tan( 1/2 ) + arc tan( 1/3 ) ) = tan( 26.565 + 18.435 ) }}}
{{{ tan( 26.565 + 18.435 ) = tan( 45 ) }}}
This is tan of the angle 45 degrees
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Now press [ tan-45 = ]
That should give you {{{ 1 }}}, which is the answer. 
It is a number, not an angle
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Note that your calculator has to be set up for angles
and not radians in order to follow this.
Your calculator shouldn't be much different
Hope this helps