Verifying Identities

5.2 Verifying Identities

Other identities can be derived from the elementary identities.

For example, an identity for ( cot A )( sin A ) can be found by replacing cot A with cos A / sin A and reducing the fraction to cos A.

Thus ( cot A )( sin A ) = cos A is an identity.

There is an important principle that must not be violated when proving identities:

Never begin by using the identity you wish to prove.

The following is a fallacious proof of the identity above.

<fallacious_proof>

( cot A )( sin A ) = cos A

( cos A / sin A )( sin A ) = cos A

cos A = cos A

</fallacious_proof>

This so-called ‘proof’ begins by using the very identity it seeks to prove. The assumption is that if we begin with some statement and go through a sequence of logical inferences and arrive at a true statement, then the original statement must have been true. The only problem is, it is possible to begin with a false statement and yet arrive at a true statement by a series of logical inferences. Thus the fact that the final statement is true implies nothing about whether the original statement is true or false. It is a common logical fallacy that only true statements imply true statements. But false statements can imply true statements. For example, consider the following fallacious ‘proof ’ that 0 = 1:

<fallacious_proof>

0 = 1

Multiplying both sides by –1 yields

( -1 )( 0) = ( -1 )( 1 ) thus

0 = - 1

Since 0 = 1 and 0 = -1, add the two equations to get

0 + 0 = 1 + ( -1 ), thus

0 = 0 which is true. Thus the original statement 0 = 1 must be true

</fallacious_proof>

This is an example of a false statement implying a true statement. These two fallacious ‘proofs’ illustrate why you cannot prove an identity if you begin by using the identity.

The following is the correct proof of the identity

<correct_proof>

( cot A )( sin A ) = ( cos A / sin A )( sin A )

( cos A / sin A )( sin A ) = cos A therefore

( cot A )( sin A ) = cos A

</correct_proof>

This may be shortened to ( cot A )( sin A ) = ( cos A / sin A )( sin A ) = cos A.

Exercise 5.2.1

Prove that sin² A = ( 1 – cos A )( 1 + cos A ).

Solution

Exercise 5.2.2

Prove that ( 1 + tan A ) / sec A = cos A + sin A