The range of a function f(x) is all the real values that f(x) can take for real values of x.

Here we have the function y = cos 2x - 4*sin x.

To find the range we have to remember that x is the same for the cos 2x as well as sin x.

y = cos 2x - 4*sin x

=> 1 - 2*(sin x)^2 - 4*sin x

sin x takes on values from -1 to + 1

For sin x = -1

y = 1 - 2 - 4 = -5

For sin x = -1

y = 1 - 2 + 4 = 3

**This gives the required range of y =cos 2x - 4*sin x as [-5 , 3]**

In other words, we'll have to identify the maximum and minimum values for y.

We'll re-write the expression of y, using just sine function.

We'll apply the double angle identity for the first term.

cos 2x = 1 - 2(sin x)^2

y = 1 - 2(sin x)^2 - 4 sin x

We'll consider the maximum value for sin x = 1:

y = 1 - 2 - 4

y = -6 + 1

y = -5

Now, we'll consider the minimum value for sin x = -1.

y = 1 - 2 + 4

y = 3

So, for a maximum value for sin x, we'll get a minimum value for y, namely y = -5. For a minimum value of sin x, we'll get a maximum of y = 3.

**The range of values of y, if y = cos2x-4sinx, is represented by the closed interval [-5 ; 3].**