102 - Topic 02 - Verifying Trigonometric Identities

170 days ago by Professor102

John Travis

Mississippi College

Verifying Identities

In proving identities, one must be careful to not logically presume the expression is true as the first step in a proof.  This often occurs when one assumes the desired identity and subsequently proceeds until ending triumphantly with something like $1 = 1$.  To complete a proof in this manner, one would need to verify explicitly that each step is always reversible.

A better way is to start with one side of a presumed identity and logically proceed until the other side pops out.  Of course, this will never work if the particular expression is not indeed an identity.  

In the Sage interactive cell below, one can type in a given proposed identity (with variable $t$) and have Sage determine whether the expression is indeed an identity.  After typing in the expressions, you may need to click out of the boxes somewhere in order for the new expressions to evaluate.

       
Verify 

Click to the left again to hide and once more to show the dynamic interactive window

# Some examples to play with # (x-1)*(x+1) = x^2-2 # (x-1)*(x+1) = x^2-1 # sec(t)*cot(t)*sin(t)/(csc(t)*tan(t)*cos(t)) = 1 # sec(t)*cot(t)*cos(t)/(csc(t)*tan(t)*sin(t)) = 1 # (sec(t)+1)*(cos(t)-1) = -tan(t)*sin(t)