Question

In any tringle abc if A=tan^1 2and B=tan^1 3 then c=?

Answer 1

`C=tan^-1 1`

Explanation:

Your question is unclear, which strange notation. I, therefore, interpret your question as follows, "If in `Delta abc`
`A=tan^-1 2` and `B=tan^-1 3` then what is `C`?"

It is also unclear if you want `C` in degrees, radians or expressed as `tan^-1`, but as the two other angles are expressed as `tan^-1`, I take it `C` should be expressed the same way. (Incidentally this shows that it is important to use correct notation, otherwise you may easily get a different answer than what you actually want.)

As a help it can be useful to create a figure first:

If so, I would go via rad to degrees and back to tan again, in the following fashioon:

`tan^-1 2` = 63.435º (see https://www.rapidtables.com/math/trigonometry/arctan.html#table) #tan^-1 3# = 71.565°

Now there are in all 180° in a triangle, so
C= 180° - (63.435º+71.565°) = 45°

As tan 45° = 1 (a well known fact, but if you are unsure, Google "tan 45 degrees", and you get the answer straight away),
your answer should be `C=tan^-1 1`

Answer 2

`C=pi/4=45^@`.

Explanation:

I hope, the Question is to find `C` of a `DeltaABC`, given that,

`A=tan^-1 2, B=tan^-1 3, i.e., tanA=2, tanB=3`.

We know that, in `DeltaABC, A+B+C=pi`.

`:. A+B=pi-C`.

`:. tan(A+B)=tan(pi-C)`.

`:. (tanA+tanB)/(1-tanAtanB)=-tanC`.

`:. (2+3)/(1-2xx3)=-tanC`.

`:. tanC=5/-5=-1, or, tanC=1`.

`:. C=pi/4=45^@`.