Oxford Entrance Exam Problem 2

Let $y=cos(x)$

Note $y$ can only range from $-1$ to $1$

Using the basic $sin^2(x)+cos^2(x)=1$ idenity we get $(-4y^{2}+4y+5)^2$

Completing the square we get $(6-4(y-\frac{1}{2})^2)^2$

Now we have two cases to check, when the inside is the largest positive number and the largest negative number.

Obviously the largest positive number is $6$ when $y=\frac{1}{2}$

The largest negative number occurs when $y=-1$ for obvious reasons. The value inside the bracket is $-3$ .

Since $6^2>(-3)^2$ the answer is $6^2=\boxed{36}$