How do you find the exact value of the expression \[\dfrac{{\tan {{25}^ \circ } + \tan {{110}^ \circ }}}{{1 - \tan {{25}^ \circ }\tan {{110}^ \circ }}}\]?

Hint:In the above question, the concept is based on the concept of trigonometry. The main approach to solve this expression is by applying trigonometric identity of tangent and also applying the tangent function behavior in a particular quadrant.

Complete step by step solution:
Trigonometric function means the function of the angle between the two sides. It tells us the relation between the angles and sides of the right-angle triangle.
The sign of all the six trigonometric functions in the first quadrant is positive since x and y coordinates are both positive. In the second quadrant the sine function and cosecant function are positive and in third only tangent and cotangent function is positive.
The fourth quadrant has only cosine and secant are positive.
So, now by applying the tangent identity i.e.,
\[\tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A \times \tan B}}\]
Here, there are two angles. The angles are A and B where A=25 and B=110.So, by substituting the values in the identity
\[\tan ({25^ \circ } + {110^ \circ }) = \tan \left( {{{135}^ \circ }} \right)\]
Now we can write the above angle in terms of 180.So, we can write it as,
\[\tan ({180^ \circ } - {45^ \circ }) = - \tan {45^ \circ }\]
According to the quadrant rules, we get negative tangent function.
\[ - \tan {45^ \circ } = - 1\]
Therefore, we get the value of the tangent as -1 according to the right-angle triangle sides measure.

Note: An important thing to note is that the angle 135 is less than 180.The angle 135 lies in the second quadrant. In the second quadrant the sign of tangent function is negative in the second quadrant hence we get a negative tangent function with negative value -1.