Proving Other Trigonometric Identities

To prove that a trigonometric equation is an identity, one typically starts by trying to show that either one side of the proposed equality can be transformed into the other, or that both sides can be transformed into the same expression.

In other words, suppose $A$ and $B$ are some trigonometric expressions and we are trying to determine if $A=B$.

We hope that both expressions will simplify to some common form $C$, as if we can show the following:

$$\begin{array}{rclcrcl} A &=& A_1 \quad && \quad B &=& B_1\\ &=& A_2 && &=& B_2\\ &=& \cdots & \textrm{and} & &=& B_3\\ &=& A_n && &=& B_4\\ &=& C && &=& \cdots\\ & & && &=& B_m\\ & & && &=& C \end{array}$$

then we will know

$$A = A_1 = A_2 = \cdots = A_n = C = B_m = \cdots = B_2 = B_1 = B$$

and thus,

$$A = B$$

That is our general "plan of attack" -- although, we might get lucky and the sequence of $A_1, A_2, \cdots$ will terminate in $B$, or the sequence $B_1, B_2, \cdots$ will terminate in $A$, which then shortens our argument a bit.

There are some basic strategies to help us get to that common form $C$ as efficiently as possible:

Showing an Equation is Not an Identity

It may be the case that in the course of trying to prove a given equation is an identity, one begins to suspect that it is not.

In such situations, one should test whether the equation's left and right sides are actually equal by plugging in some values for the variables it contains. Remember, one only needs a single counter-example to prove an equation is not an identity.

If however, one tests a particular value (or set of values) and the left and right sides of the given equation agree in value, that particular test is inconclusive -- and a decision must be made whether to continue the search for a counter-example and test additional values, or to return to trying to prove the given equation is an identity.

To have the best chance of selecting values that will show a given equation is not an identity, one should keep the following in mind: