How do you simplify: $ {\sin ^2}\left( {2x} \right) - {\cos ^2}\left( {2x} \right) $ .

Hint: The given question deals with basic simplification of trigonometric functions by using some of the simple trigonometric formulae such as double angle formula of cosine: $ \cos \left( {2\theta } \right) = {\cos ^2}\theta - {\sin ^2}\theta $ . Basic algebraic rules and trigonometric identities are to be kept in mind while doing simplification in the given problem.

Complete step-by-step solution:
In the given problem, we have to simplify the $ {\sin ^2}\left( {2x} \right) - {\cos ^2}\left( {2x} \right) $ .
So, $ {\sin ^2}\left( {2x} \right) - {\cos ^2}\left( {2x} \right) $ .
First taking the negative sign out of the bracket, we get,
 $ \Rightarrow - \left[ {{{\cos }^2}\left( {2x} \right) - {{\sin }^2}\left( {2x} \right)} \right] $
Using $ \cos \left( {2\theta } \right) = {\cos ^2}\theta - {\sin ^2}\theta $ ,
 $ \Rightarrow - \cos \left( {4x} \right) $
So, we have simplified the given trigonometric expression $ {\sin ^2}\left( {2x} \right) - {\cos ^2}\left( {2x} \right) $ as $ - \cos \left( {4x} \right) $ using algebraic rules and trigonometric formula.

Additional information:
There are $ 6 $ trigonometric functions, namely: $ \sin (x) $ , $ \cos (x) $ , $ \tan (x) $ , $ \cos ec(x) $ , $ \sec (x) $ and \[\cot \left( x \right)\]. Also, $ \cos ec(x) $ , $ \sec (x) $ and \[\cot \left( x \right)\]are the reciprocals of $ \sin (x) $ , $ \cos (x) $ and $ \tan (x) $ respectively. Given problem deals with Trigonometric functions. For solving such problems, trigonometric formulae should be remembered by heart such as the double angle formula for cosine: $ \cos \left( {2\theta } \right) = {\cos ^2}\theta - {\sin ^2}\theta $ .

Note: Besides these simple trigonometric formulae, trigonometric identities are also of significant use in such types of questions where we have to simplify trigonometric expressions with help of basic knowledge of algebraic rules and operations. However, questions involving this type of simplification of trigonometric ratios may also have multiple interconvertible answers. Trigonometric functions are also called Circular functions. Trigonometric functions are the functions that relate an angle of a right angled triangle to the ratio of two side lengths.