2.4: Introduction to Proofs in Trigonometry

Simplifying Expressions Involving Trigonometric Functions

Recall that an algebraic expression (or mathematical expression) is a combination of symbols that are mathematically "well-formed." The mathematical symbols can include numbers (constants), variables, operations (e.g., addition, subtraction, multiplication, etc.), functions, brackets, and other grouping symbols to help determine the order of operations. The one symbol that is always missing from an expression is the equals sign (\(=\)).

When we simplify an algebraic expression, we obtain a new expression that has the same values as the old one, but is easier to work with. For example, we can apply the Distributive Law and combine like terms to simplify\[\begin{array}{rcl} 2x(x-6)+3(x+2) & = & 2x^2-12x+3x+6 \\[6pt] & = & 2x^2-9x+6 \\[6pt] \end{array}\nonumber\]The new expression is equivalent to the old one, that is, the expressions have the same value when we evaluate them at any value of \(x\). For instance, you can check that, at \(x=3\), the expressions \(2x(x-6)+3(x+2)\) and \(2x^2-9x+6\) become\[\begin{array}{rcccl} 2(3)(3-6)+3(3+2) & = & 6(-3)+3(5) & = & -3 \\[6pt] 2(3)^2-9(3)+6 & = & 18-27+6 & = & -3 \\[6pt] \end{array}\nonumber\]To simplify an expression containing trigonometric functions, we treat each function as a single variable. Compare the two calculations below:\[\begin{array}{ccccc} 8xy & - & 6xy & = & 2xy \\[6pt] 8\cos\left(\theta\right)\sin\left(\theta\right) & - & 6\cos\left(\theta\right)\sin\left(\theta\right) & = & 2\cos\left(\theta\right)\sin\left(\theta\right) \\[6pt] \end{array}\nonumber\]Both calculations are examples of combining like terms. In the second calculation, we treat \(\cos\left(\theta\right)\) and \(\sin\left(\theta\right)\) as variables, just as we treat \(x\) and \(y\) in the first calculation.

In Trigonometry, we are often tasked with simplifying expressions involving trigonometric functions. In doing so, we will use many of our skills from Algebra (e.g., simplifying compound rational expressions, factoring, distributing, etc.) in combination with the identities we have recently discovered (along with those we will soon discover). One strategy for simplifying a trigonometric expression is to reduce the number of different trigonometric functions involved. The following example showcases this process using the Ratio Identities.

It is often easier to know how a trigonometric expression (an expression involving trigonometric functions) will simplify once you try simplification techniques. In Example \(\PageIndex{1a}\), most students new to Trigonometry would likely never have looked at \(\cos\left(\theta\right)\tan\left(\theta\right)+\sin\left(\theta\right)\) and thought, "Hey, I bet that simplifies down to something nice... like \(2\sin\left(\theta\right)\)." Luckily, as you move forward in Trigonometry (and mathematics), you develop an intuition for when an expression can be simplified; however, cultivating this intuition takes time and experimentation.

Rewriting Trigonometric Expressions

It is often necessary, especially in Calculus, to rewrite a trigonometric expression in terms of a single trigonometric function. To do so, we must use identities.

If we graph the original expression from Example \(\PageIndex{2a}\) as \(y_1=\sin\left(x\right)\cos^2\left(x\right)\) and our resulting equivalent expression as \(y_2=\sin\left(x\right)-\sin^3\left(x\right)\), we see that they have the same graph, as shown in Figure \(\PageIndex{1}\) below.

This should convince us that \(\sin\left(\theta\right)\cos^2\left(\theta\right)\) truly is equivalent to \(\sin\left(x\right)-\sin^3\left(x\right)\).

The results of Example \(\PageIndex{2}\) can be thought of as two new identities:\[\sin\left(\theta\right)\cos^2\left(\theta\right)=\sin\left(\theta\right)-\sin^3\left(\theta\right) \quad \text{and} \quad \cot\left(\theta\right) = \pm\dfrac{\cos\left(\theta\right)}{\sqrt{1-\cos^2\left(\theta\right)}},\nonumber\]however, before you get too concerned with having to memorize these as two more identities, let's be clear:

Unless formally stated as a theorem, there is no need to memorize the hundreds of identities we will create, prove, or encounter in Trigonometry.

This means that the only identities you are responsible for memorizing (so far) are the Reciprocal, Ratio, and Pythagorean Identities.

The following example is extremely useful in Calculus II (Integral Calculus).

Our solution to Example \(\PageIndex{3}\) needs some clarification to ensure you understand what happened. Most of the work should be understandable; however, two steps might throw you off.

First, from the Pythagorean Identities, we used the fact that\[1+\tan^2\left(\theta\right) = \sec^2\left(\theta\right);\nonumber\]however, we modified this identity slightly by subtracting 1 from both sides to get\[\tan^2\left(\theta\right) = \sec^2\left(\theta\right)-1.\nonumber\]The implication of that subtle modification cannot be overstated.

Being comfortable with the available identities and willing to manipulate them as needed will play a critical role in your success in Trigonometry.

The second item that needs our attention is the mathematical equivalence\[\sqrt{\tan^2\left(\theta\right)} = \left| \tan\left(\theta\right)\right|.\nonumber\]A lot of students forget about the absolute values. Let's focus on what is happening here.

Suppose a friend of yours is claiming that the equation \(\sqrt{x^2}=x\) is an identity (this is the same as someone saying \(\sqrt{\tan^2\left(\theta\right)} = \tan\left(\theta\right)\)). As an astute mathematics student, you know that even though the equation is true for all positive values of \(x\), it is false for negative values of \(x\). For example, if \(x=-3\), then\[\sqrt{x^2}=\sqrt{(-3)^2}=\sqrt{9}=3\nonumber\]so \(\sqrt{x^2} \neq x\). The radical symbol \(\sqrt{ }\) stands for the principal square root—that is, the nonnegative square root. Therefore, the left side of the equation, \(\sqrt{x^2}\), is never negative. Thus, \(\sqrt{x^2}\) cannot equal \(x\) when \(x\) is a negative number. The equation is false for \(x<0\).

One way to see that \(\sqrt{x^2}\) and \(x\) are not equivalent is to compare the graphs of \(Y_1=\sqrt{x^2}\) and \(Y_2=x\), shown in Figure \(\PageIndex{2}\) below. You can see that \(\sqrt{x^2}\) and \(x\) do not have the same value for \(x<0\).

Coming back to the last step in the solution of Example \(\PageIndex{3}\), we should now feel comfortable saying that \(\sqrt{\tan^2\left(\theta\right)} \neq \tan\left(\theta\right)\) and we should be okay with saying\[\sqrt{\tan^2\left(\theta\right)} = \left|\tan\left(\theta\right)\right|.\nonumber\]

Checking the Plausibility of Identities Graphically

Before jumping into how to rigorously prove a trigonometric identity, let's focus on ways to show that a claimed identity is not an identity.

From the discussion after Example \(\PageIndex{3}\), we can see that, to check whether an equation might be an identity, we can compare graphs of \(Y_1=\)(left side of the equation) and \(Y_2=\) (right side of the equation). If the two graphs are identical, it is plausible that the equation is an identity. If the two graphs differ, the equation is not an identity.

Read that last paragraph again.

You cannot use a graph to prove an equation is an identity; however, you can use a graph to demonstrate it is not an identity.

This is crucial to understand. The following example provides some clarity.

Example \(\PageIndex{4}\) has a few cautionary tales.

Proving Identities

We have proved a trigonometric identity when we show that one trigonometric expression is equivalent to another. In Example \(\PageIndex{1a}\) we proved that the equation\[\cos\left(\theta\right)\tan\left(\theta\right)+\sin\left(\theta\right)=2\sin\left(\theta\right)\nonumber\]is an identity; it is valid for all values of \(\theta\) (as long as the tangent function is defined).

A common strategy for proving an identity is to transform one side of the equation using equivalent expressions until it is identical to the other side. To help us choose the transformations at each step of the proof, we try to match the algebraic form of the final expression.

When you write out the proof of an identity, your goal is to transform the expression on one side of the identity into the expression on the other, showing one step of the calculation on each line of your proof. You can justify each step to the right of the calculation. The proof of the identity in the previous example would look like this:\[\begin{array}{rclcr} \text{LHS} & = & 1+\tan^2\left(t\right) & & \\[6pt] & = & 1+\left(\dfrac{\sin\left(t\right)}{\cos\left(t\right)}\right)^2 & \quad & (\text{Ratio Identities}) \\[6pt] & = & 1+\dfrac{\sin^2\left(t\right)}{\cos^2\left(t\right)} & \quad & (\text{Laws of Exponents}) \\[6pt] & = & \dfrac{\cos^2\left(t\right)}{\cos^2\left(t\right)}+\dfrac{\sin^2\left(t\right)}{\cos^2\left(t\right)} & \quad & (\text{Get a common denominator to combine fractions}) \\[6pt] & = & \dfrac{\cos^2\left(t\right)+\sin^2\left(t\right)}{\cos^2\left(t\right)} & \quad & (\text{adding fractions}) \\[6pt] & = & \dfrac{1}{\cos^2\left(t\right)} & \quad & (\text{Pythagorean Identity}) \\[6pt] & = & \text{RHS} & & \\[6pt] \end{array}\nonumber\]For now, we will focus on transforming the left side of an equation into the right side; however, as we move forward in Trigonometry, we will adopt the rule of thumb of transforming the more "complicated" side of the equation into the other side.

Before we leave this section, it's important to note that you will be proving many identities in Trigonometry. It takes PRACTICE! We have introduced the idea of a proof early so that we spend a good deal of time sharpening those skills and giving advice on tactics as we move forward. For now, the best advice is to do all the proofs in the homework section—even if your instructor does not assign them.