If $$x\cos \theta + y\sin \theta = z$$, then what is the value of $${(x\sin \theta - y\cos \theta )^2}$$ ?

If $$\sin 18^\circ = {{\sqrt 5 - 1} \over 4}$$, then what is the value of sin81$$^\circ$$ ?

What is $${{1 - \tan 2^\circ \cot 62^\circ } \over {\tan 152^\circ - \cot 88^\circ }}$$ equal to?

Consider the function $$f(\theta ) = 4({\sin ^2}\theta + {\cos ^4}\theta )$$

What is the maximum value of the function $$f(\theta )$$ ?