Pythagorean Identities and Proof
Neil Trivedi
Teacher
Pythagorean Identities and Proof
The new Pythagorean identities are built on the equation , which we are already familiar with.
Identities:
Proof:
We prove both identities with .
Identity 1: Divide each term in the identity by .
Identity 2: Divide each term in the identity by .
Example 1:
Prove
where and .
Step 1: Combine the fractions on the LHS.
First expand the brackets in the numerator like normal and the denominator using the difference of two squares principle (DOTS). Then simplify the numerator and collect like terms.
Step 2: Convert using Pythagorean Identities.
Knowing that , we can square root both sides to obtain:
So,
No answer provided.
Example 2:
Prove the following identities:
a)
Single Step: Convert the expressions to sine and cosine and simplify the LHS.
b)
Single: Combine the fractions and simplify the LHS.
Expanding the brackets in the denominator using DOTS and collecting the like terms in the numerator,
Using our new trigonometric identity
c)
Step 1: Make use of DOTS, where , on the LHS and simplify.
Using the identity ,
Step 2: Convert the expressions to sine and cosine and simplify.
No answer provided.
Practice Questions