Chapter 6.5, Problem 53AYU

Exploration Graph

$y\text{ = }\tan x\text{ and }y\text{ = }-\cot \left(x\text{ + }\frac{\pi }{2}\right)$

Do you think that $\tan x\text{ = }-\cot \left(x\text{ + }\frac{\pi }{2}\right)$?

To determine

To find: Do you think that $\tan x=-\cot \left(x+\frac{\pi }{2}\right)$?

Answer to Problem 53AYU

Solution:

The two graphs align exactly over one period and hence all periods which indicates that these are in fact the same functions and hence one should think that $\tan x=-cot\left(x+\frac{\pi }{2}\right)$.

Explanation of Solution

Given:

$y=\tan x\text{ and }y=-\cot \left(x+\frac{\pi }{2}\right)$

Calculation:

The graph of the tangent function.

The graph of the inverted and shifted cotangent function.

The two graphs align exactly over one period and hence all periods which indicates that these are in fact the same functions and hence one should think that $\tan x=-\cot \left(x+\frac{\pi }{2}\right)$. If you were to graph these two functions on the same axes, the graphs would superimpose exactly, and you could not distinguish one from the other. This is like a graphical proof of identity.