Find the value of: $\sin ^{ 2 }{ 30 } \cos ^{ 2 }{ 45 } +4\tan ^{ 2 }{ 30 } +\cfrac { 1 }{ 2 } \sin ^{ 2 }{ 90 } +\cfrac { 1 }{ 2 } \cot ^{ 2 }{ 60 } $

Substitute the standard trigonometric values into the expression: sin 30 equals one half, cos 45 equals one divided by the square root of 2, tan 30 equals one divided by the square root of 3, sin 90 equals 1, and cot 60 equals one divided by the square root of 3. The first term becomes one fourth times one half, which is one eighth, and the second term becomes 4 times one third, which is four thirds. The third term is one half, and the fourth term is one half times one third, which is one sixth. Finding a common denominator of 24 for all terms yields 3 divided by 24 plus 32 divided by 24 plus 12 divided by 24 plus 4 divided by 24, which adds up to 51 divided by 24 and simplifies to 17 divided by 8.

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