Trigonometric Identities

Last Thursday and Friday we've learned about Trigonometric Identities.

A Trigonometric Function is by definition an equation that involves at least one trigonometric function of a variable. Such an equation is called a Trigonometric Identity if is is true for all values of the variable for which both sides of the equation are defined.

Verifying an identity, or proving that a given equation is an identity, is to show that the left hand side (LHS) of the equation is identical to the right hand side(RHS).

There Basic Identities, which are:




Pythagorean Identities:




*Note that you could always switch things from one side to the other side to create other identities.

Example 1:




Example 2:




Verifying Identities means to prove that LHS = RHS

Method One - use the fundamental identities to change one side (usually the more difficult one) of the equation into the form of the other side.

For Example:




Step 1: State Side
For this equation, it is the Right Hand Side.




Step 2: Do the work - change info




























Step 3: State = other side
=



RHS = LHS



Method Two - use fundamental identities to change each side (split by a vertical line) of the equation until the same form is obtained on each side. Once sides are equal state that LHS = RHS.

For Example:


Step 1: Simplify Both Sides

LHS =

Which equals to...


And that's all you can pretty much do on the LHS of the function.

RHS =




Find the common denominator...






Then add up the denominator...




Flip then Multiply...






And you're left with...






Then multiply the denominator by its conjugate...


Conjugation - when the denominator is 1 -/+ sinx, cosx, tanx etc. The function is multiplied by the exact denominator but with the opposite sign.





Multiply...










Which Equals to...




Cancel ...



Then you're left with...




Which equals to the Left Hand Side (LHS)...
= LHS


So you can state that...
RHS = LHS