prove that the value of cosx + secx can never be 3/2.
First make a sketch, here is Wolfram's version
http://www.wolframalpha.com/input/?i=y+%3D+cosx+%2B+1%2Fcosx+for+-%CF%80+to+%2B%CF%80
The vertical lines are asymptotes, and thus will not yield any values. You can see that the min value is
appr 2
Here is a larger picture, showing that it is periodic
http://www.wolframalpha.com/input/?i=y+%3D+cosx+%2B+1%2Fcosx+
let y = cosx + secx
dy/dx = -sinx + secx tanx
= 0 for a max or min
secx tanx = sinx
1/cosx * sinx/cox = sinx
sinx/cos^2 x = sinx
times cos^2 x
sinx = sinx(cos^2 x)
sinx - sinx(cos^2x) = 0
sinx(1 - cos^2 x) = 0
sinx(sin^2 x) = 0
sin^3 x = 0
sinx = 0
sinx x = 0
x = 0, π, 2π, 3π, ...
so let's look at some of the y values
if x = 0,
y = 1 + 1/1 = 2 , ahh, so it was exactly 2
if x = π
y = -1 -1 = -2
So that seems to be our max and min at its corresponding loop
that is, in the main loop between -π and +π, the min is 2
so there are no values of the function that fall between
-2 and +2
and since 3/2 lies in that range, we have proven the case.
To prove that the value of cos(x) + sec(x) can never be 3/2, we can begin by manipulating the given expression.
First, recall the reciprocal identity for sec(x):
sec(x) = 1/cos(x)
Now, let's substitute this into the original expression:
cos(x) + sec(x) = cos(x) + 1/cos(x)
Next, let's simplify this expression by finding the common denominator:
cos(x) + 1/cos(x) = (cos^2(x) + 1) / cos(x)
Now, we have:
(cos^2(x) + 1) / cos(x)
To further simplify, we observe that cos^2(x) + 1 is equivalent to sin^2(x). Using the Pythagorean identity, sin^2(x) + cos^2(x) = 1:
cos(x) + 1/cos(x) = sin^2(x) / cos(x)
Now, let's focus on the numerator, sin^2(x). We know that sin^2(x) is always less than or equal to 1. Therefore, sin^2(x)/cos(x) will always be less than or equal to cos(x).
Now, recall that the range of the cosine function is [-1, 1]. Thus, cos(x) can take on any value between -1 and 1. Since sin^2(x)/cos(x) is always less than or equal to cos(x), it follows that sin^2(x)/cos(x) can never be greater than or equal to 1, let alone equal to 3/2.
Hence, we have proved that the value of cos(x) + sec(x) can never be 3/2.