sin^2 x + 4 sinx +3 3 + sinx
-------------------------- = -------------------
cos^2 x 1 - sinx
factor the numerator
(sinx +3) (sinx+1) 3 + sinx
-------------------------- = -------------------
cos^2 x 1 - sinx
cos^2 = 1-sin^2x
(sinx +3) (sinx+1) 3 + sinx
-------------------------- = -------------------
1- sin^2x 1 - sinx
factor the denominator
(sinx +3) (sinx+1) 3 + sinx
-------------------------- = -------------------
(1-sinx ) (1+sinx) 1 - sinx
cancel the common term (1+sinx) and (sinx +1)
(sinx +3) 3 + sinx
-------------------------- = -------------------
(1-sinx ) 1 - sinx
reorder the first term
3+sinx 3 + sinx
-------------------------- = -------------------
(1-sinx ) 1 - sinx
Given that 
, we have 
, so that
Take the derivative and find the critical points of 
:
Take the second derivative and evaluate it at the critical point:
Since 
is positive for all 
, the critical point is a minimum.
At the critical point, we get the minimum value 
.
The answer is that it is positive
12 ) 18
13) 54
14) 28/54 (8weeks x 7 days) = 1/2
18/24 = 3/4
therefore rates not equal
15) 0 and 0 ;1 and 4; 2 and 8;3 and 12
