## 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