The function  $f\left(x\right)$ is defined by   $f\left(x\right)={\cos }^{4}x+K{\cos }^{2}2x+{\sin }^{4}x$, where K is a constant. If the function  $f\left(x\right)$ is a constant function, the value of K is : 

Your first impulse like mine might be to complete the square to get the expression       ${({\cos }^{2}x+{\sin }^{2}x)}^2$ somewhere in the problem. While this works, it is easier to substitute in  two values for x and force them to be equal, since the function is to be constant. Any two values will work but, we want to use ones for which we know the sine and cosine. All multiples of   $\frac{\pi }{2}$result in 1+K. so we need to put in one value that is not a multiple of  $\frac{\pi }{2}$. Try $\frac{\pi }{4}$  this gives us   

$f\left(\frac{\pi }{4}\right)={\cos }^4\left(\frac{\pi }{4}\right)+K{\cos }^2\left(\frac{\pi }{2}\right)+{\sin }^4\left(\frac{\pi }{4}\right)=\frac{1}{4}+K\cdot 0+\frac{1}{4}=\frac{1}{2}$
So now we know that 
    $1+K=\frac{1}{2},SoK=\frac{1}{2}$
This does in fact give us the function
    $f\left(x\right)={({\cos }^{2}x+{\sin }^{2}x)}^2-\frac{1}{2}=\frac{1}{2}$