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5.2 Verifying Trigonometric Identities

Discovering other identities

Other trigonometric 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 \(\dfrac{\cos A}{\sin A}\) and simplifing to \(\cos A\).

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

A caveat

Never begin a proof by assuming the truth of that which you are attempting to prove.

The following is an invalid proof of the identity above.

\( \begin{eqnarray*} \cot A\sin A &=& \cos A\\[12pt] \dfrac{\cot A\sin A}{\sin A}&=&\dfrac{\cos A}{\sin A}\\[12pt] \cot A &=& \cot A \end{eqnarray*} \)

This so-called ‘proof’ begins by using the very identity it seeks to prove. The presumption 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. But this presumption is false. It is possible to begin with a false statement and yet arrive at a true statement by a series of logical inferences. 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 invalid proof that \(0=1\).

\(0 = 1\)

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.

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.

Exercise 5.2.1

Verify that \(\sin^2 A = ( 1 – \cos A )( 1 + \cos A )\).

See Solution

Exercise 5.2.2

Verify that \(\dfrac{1+\tan A}{\sec A} = \cos A + \sin A\)

See Solution

Exercise 5.2.3

Verify that \(\dfrac{1}{\sec A + \tan A}= \sec A – \tan A\)

See Solution

Exercise 5.2.4

Verify that \(\cos ( \frac{\pi}{2} – A ) \sec A = \cot ( \frac{\pi}{2} – A )\)

See Solution