Proving Identities in Trigonometry ( Add Maths)

Hi students, so you are wondering how to prove identities when even memorizing the formulaes are already a big problem? Fret not.

First, you must recognize that proving identities is not the same as when you are required to SOLVE the equation. In solving you must find what the value of the variable ( say, x or y or theta or whatever). On the other hand, in proving identities you have to realize that it is already an equation or a statement which is true.

In proving identities, use logical steps to show that whatever on the Left Hand Side is equal or can be transformed to the anything you are seeing on the Right Hand Side of the equation.

Secondly, DO NOT work on both sides of the equation simultaneously. In solving equation you can do this, but in proving identities, this is incorrect. If you do this you are merely proving that the entire equation is just, well, true.You are supposed to prove one thing is equal to another.


As an example,

(sinx - sin x cos^2 x )/ cos x - cos^3 x   = tan x

someone tried this..

tan x(sinx - sin x cos^2 x ) = cos x - cos^3 x

a big NO. this is incorrect in proving identities.



You should either choose the part of the equation on the left hand side and prove it to be equivalent to the other part of the equation which is on the right hand side or vice versa.

By the way, since we are going to use either side of the equation, we will have to decide to use some notations. As mentioned, we use Left hand side, which can be annotated as LHS or right hand side as RHS).

LHS:
(sinx - sin x cos^2 x )/ cos x - cos^3 x       
= [sin x ( 1 - cos^2 x)}/ [cos x (1- cos^2 x)]     (Note: factor the top and the bottom and cancel out (1- cos^2 x))
= tan x                                                                           
( RHS)

 Easy?

Let's try another one.

Prove that 
tan x + cot x = 2 cosec 2x


Don't look at the answer until you have tried.


Answer:

LHS: tan x + cot x = (sin x / cos x ) + (cos x/ sin x)                 [note: change to sin and cos]
                             = (sin^2 x  + cos^2 x ) / (cos x sin x)         [ create common denominator]
                            =  1 / (cos x sin x)                                       [ yes, (sin^2 x  + cos^2 x ) is equal to 1]
                            =  1 / (sin 2x/2)                      [ now, sin 2x = (2 sin x cos x ) so, what is sin x cos x?]
                            =  2/ sin 2x                                   [Tidy up, 2 goes up]
                            =  2 cosec x                                  Tadaaa..
                            = RHS

Okay till our next posting. Have a good weekend people. Have a good rest.