Evaluate the given trigonometric equation ${\cos ^2}\left( {67} \right) - {\sin ^2}\left( {23} \right)$.

Hint – In this question use the concept that $\sin \left( {90 - \theta } \right) = \cos \theta $, to convert ${\sin ^2}\left( {23} \right)$ into ${\cos ^2}\left( {67} \right)$. This will evaluate the trigonometric relation.

Complete step-by-step answer:
Given trigonometric equation is
${\cos ^2}\left( {67} \right) - {\sin ^2}\left( {23} \right)$
Above equation is also written as
\[ \Rightarrow {\cos ^2}\left( {67} \right) - {\left( {\sin \left( {90 - 67} \right)} \right)^2}\]
Now as we know that $\sin \left( {90 - \theta } \right) = \cos \theta $ so use this property in above equation we have,
$ \Rightarrow {\cos ^2}\left( {67} \right) - {\cos ^2}\left( {67} \right)$
Now as we see both terms cancel out so the solution is zero (0).
$ \Rightarrow {\cos ^2}\left( {67} \right) - {\sin ^2}\left( {23} \right) = 0$
So this is the required answer.

Note – Such types of problems are solemnly based upon the application of trigonometric identities. Some of the important identities involve $\cos ({180^0} - x) = - \cos x,{\text{ cos(18}}{{\text{0}}^0} + x) = - \cos x$. It is advised to remember these identities as it helps saving a lot of time while solving problems of this kind.