Without using the trigonometric tables, evaluate ${\sin ^2}{34^ \circ } + {\sin ^2}{56^ \circ } + 2\tan {18^ \circ }\tan {72^ \circ } - {\cot ^2}{30^ \circ }$.

Hint: The given problem requires us to simplify the given trigonometric expression. The question requires thorough knowledge of trigonometric functions, formulae and identities. The question describes the wide ranging applications of trigonometric identities and formulae. We must keep in mind the trigonometric identities and formulae while solving such questions.

Complete step-by-step answer:
In the given question, we are required to evaluate the value of the trigonometric expression ${\sin ^2}{34^ \circ } + {\sin ^2}{56^ \circ } + 2\tan {18^ \circ }\tan {72^ \circ } - {\cot ^2}{30^ \circ }$ using the basic concepts of trigonometry and identities.
So, we have, ${\sin ^2}{34^ \circ } + {\sin ^2}{56^ \circ } + 2\tan {18^ \circ }\tan {72^ \circ } - {\cot ^2}{30^ \circ }$
Firstly, we simplify the given trigonometric expression using the trigonometric formula \[\cos \left( {{{90}^ \circ } - x} \right) = \sin x\]. So, we get,
\[ \Rightarrow {\sin ^2}{34^ \circ } + {\cos ^2}\left( {{{90}^ \circ } - {{56}^ \circ }} \right) + 2\tan {18^ \circ }\tan {72^ \circ } - {\cot ^2}{30^ \circ }\]
Now, we also know that tangent and cotangent are complementary functions of each other. So, we have, \[\tan \left( {{{90}^ \circ } - x} \right) = \cot x\]. Simplifying the expression, we get,
\[ \Rightarrow {\sin ^2}{34^ \circ } + {\cos ^2}{34^ \circ } + 2\tan {18^ \circ }\cot \left( {{{90}^ \circ } - {{72}^ \circ }} \right) - {\cot ^2}{30^ \circ }\]
\[ \Rightarrow {\sin ^2}{34^ \circ } + {\cos ^2}{34^ \circ } + 2\tan {18^ \circ }\cot \left( {{{18}^ \circ }} \right) - {\cot ^2}{30^ \circ }\]
Now, we know that the tangent and cotangent are reciprocal trigonometric functions that means $\cot x\tan x = 1$. So, we get the value of $\cot {18^ \circ }\tan {18^ \circ } = 1$
Hence, we get,
\[ \Rightarrow {\sin ^2}{34^ \circ } + {\cos ^2}{34^ \circ } + 2\left( 1 \right) - {\cot ^2}{30^ \circ }\]
Using the trigonometric identity ${\sin ^2}x + {\cos ^2}x = 1$, we get,
\[ \Rightarrow 1 + 2 - {\cot ^2}{30^ \circ }\]
We also know that the value of $\cot {30^ \circ }$ is $\sqrt 3 $. So, substituting the value of $\cot {30^ \circ }$ in the trigonometric expression, we get,
\[ \Rightarrow 3 - {\left( {\sqrt 3 } \right)^2}\]
Simplifying the expression further, we get,
\[ \Rightarrow 3 - 3 = 0\]
So, we get the value of trigonometric expression ${\sin ^2}{34^ \circ } + {\sin ^2}{56^ \circ } + 2\tan {18^ \circ }\tan {72^ \circ } - {\cot ^2}{30^ \circ }$ as zero.
So, the correct answer is “0”.

Note: There are six trigonometric ratios: $\sin \theta $, $\cos \theta $, $\tan \theta $, $\cos ec\theta $, $\sec \theta $and $\cot \theta $. Basic trigonometric identities include ${\sin ^2}\theta + {\cos ^2}\theta = 1$, ${\sec ^2}\theta = {\tan ^2}\theta + 1$ and $\cos e{c^2}\theta = {\cot ^2}\theta + 1$. These identities are of vital importance for solving any question involving trigonometric functions and identities. All the trigonometric ratios can be converted into each other using the simple trigonometric identities listed above. The given problem involves the use of trigonometric formulae and identities. Such questions require thorough knowledge of trigonometric conversions and ratios. Algebraic operations and rules like transposition rule come into significant use while solving such problems.