TRIGONOMETRY Proving Trigonometric Identities

REVIEW Quotient Identities Reciprocal Identities Pythagorean Identities

Let’s start by working on the left side of the equation….

Rewrite the terms inside the second parenthesis by using the quotient identities

Simplify

To add the fractions inside the parenthesis, you must multiply by one to get common denominators

Now that you have the common denominators, add the numerators

Simplify

Since the left side of the equation is the same as the right side, you’ve successfully proven the identity!

On to the next problem….

Let’s start by working on the left side of the equation….

We’ll factor the terms using the difference of two perfect squares technique

Using the Pythagorean Identities the second set of parenthesis can be simplified

Since the left side of the equation is the same as the right side, you’ve successfully proven the identity!

On to the next problem….

Let’s start by working on the right side of the equation….

Multiply by 1 in the form of the conjugate of the denominator

Now, let’s distribute in the numerator….

… and simplify the denominator

‘ Split’ the fraction and simplify

Use the Quotient and Reciprocal Identities to rewrite the fractions

And then by using the commutative property of addition…

… you’ve successfully proven the identity!

One more….

Let’s work on the left side of the equation…

Multiply each fraction by one to get the LCD

Now that the fractions have a common denominator, you can add the numerators

Simplify the numerator

Use the Pythagorean Identity to rewrite the denominator

Multiply the fraction by the constant

Use the Reciprocal Identity to rewrite the fraction to equal the expression on the right side of the equation