Hi thereddevils

As Mr. F has said : subs. the endpoints of the domain

When x = 0 --------------------> cos x = 1 and sin x = 0

When x = 90 degrees -------> cos x = 0 and sin x = 1

So, in this case, the maximum value of sin and cos is 1 and minimum value is 0.

When you want to find the maximum value of $\displaystyle \frac{1+cosx}{1+2sinx+2cosx}$, you make the value of the numerator as big as possible and the denominator as small as possible.

Because the max. value of cos is 1, then you take x = 0 to optimize the numerator. For x = 0, the value of sin = 0, so you'll get :

$\displaystyle \frac{1+cosx}{1+2sinx+2cosx}$

$\displaystyle =\frac{1+\cos (0)}{1+2\sin(0)+2\cos(0)}$

$\displaystyle =\frac{1+1}{1+2*0+2*1}$

$\displaystyle =\frac{2}{3}$

To find the min. value, you make the numerator as small as possible and the denominator as big as possible. Give it a try