Understanding the Pythagorean Identities



These are derived from the Pythagorean Theorem and the unit circle. And they are very useful when for manipulating and solving equations.
Before you can use these to solve equations, you must understand how to manipulate them.
Examples:
1) Simplify

Since we know that
we can replace
with 1:

This is good, but we can go even further. If
then we can rearrange this identity by moving the 1 to the other side:
and then multiplying both sides by -1:
. Thus, our expression actually equals:

Being able to manipulate expressions will be helpful when solving more complex equations.



This is good, but we can go even further. If




Being able to manipulate expressions will be helpful when solving more complex equations.
2) Rewrite

We can replace 1 with
:

Now we can add like terms.
. Thus our equation is now:


Now we can add like terms.


3) Simplify

First, since
, let's replace 

We know that
, but there is no 1 to replace. However, if we subtract 5 from both sides then there will be.

Now we can substitute
for the 1







We know that


Now we can substitute



Practice: Use the Pythagorean Identities to rewrite the following expressions as instructed:
1) Simplify

2) Simplify

3) Rewrite this expression to include sin instead of cos:

4) Rewrite this expression to include cot instead of csc:

5) Simplify

Answers (Equivalent answers are also possible)
1)





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