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Trigonometry Through Wayfinding and Navigation Across the Pacific

Kamuela E Yong

Section 3.1 Trigonometric Identities

In this chapter, we explore trigonometric identities and formulas, essential tools that enable us to algebraically manipulate and solve complex trigonometric equations. These identities and formulas enable us to analyze expressions in various forms, often simplifying complex expressions into ones that are easily solvable and interpretable. By doing so, we increase our ability to accurately model the world around us.

Subsection 3.1.1 Fundamental Trigonometric Identities

An identity in mathematics is an equation that remains true for every valid values of its variables. We begin by reviewing some of the basic trigonometric identities from Chapter 1, collectively known as the fundamental trigonometric identities.

Definition 3.1.1.

The fundamental trigonometric identities are
  1. Reciprocal Identities (Definition 1.4.2)
    \begin{align*} \sin\theta\amp =\dfrac{1}{\csc\theta} \amp \cos\theta\amp =\dfrac{1}{\sec\theta} \amp \tan\theta\amp =\dfrac{1}{\cot\theta}\\ \csc\theta\amp =\dfrac{1}{\sin\theta} \amp \sec\theta\amp =\dfrac{1}{\cos\theta} \amp \cot\theta\amp =\dfrac{1}{\tan\theta} \end{align*}
  2. Quotient Identities (Definition 1.4.3)
    \begin{align*} \tan\theta \amp =\dfrac{\sin\theta}{\cos\theta} \amp \cot\theta\amp =\dfrac{\cos\theta}{\sin\theta} \end{align*}
  3. Pythagorean Identities (Definition 1.5.20)
    \begin{align*} \sin^2\theta+\cos^2\theta \amp = 1 \\ 1+\tan^2\theta \amp = \sec^2\theta\\ 1+\cot^2\theta \amp = \csc^2\theta \end{align*}
  4. Odd-Even Identities (Definition 1.5.22)
    The cosine and secant functions are even
    \begin{align*} \cos(-\theta)\amp =\cos\theta \amp \sec(-\theta)\amp =\sec\theta \end{align*}
    The sine, cosecant, tangent, and cotangent functions are odd
    \begin{align*} \sin(-\theta)\amp =-\sin\theta \amp \csc(-\theta)\amp =-\csc(\theta)\\ \tan(-\theta)\amp =-\tan\theta \amp \cot(-\theta)\amp =-\cot(\theta) \end{align*}
  5. Cofunction Identities (Definition 1.4.7)
    \begin{align*} \sin\theta\amp =\cos\left(\frac{\pi}{2}-\theta\right), \amp \cos\theta\amp =\sin\left(\frac{\pi}{2}-\theta\right)\\ \tan\theta\amp =\cot\left(\frac{\pi}{2}-\theta\right), \amp \cot\theta\amp =\tan\left(\frac{\pi}{2}-\theta\right)\\ \sec\theta\amp =\csc\left(\frac{\pi}{2}-\theta\right), \amp \csc\theta\amp =\sec\left(\frac{\pi}{2}-\theta\right) \end{align*}

Subsection 3.1.2 Simplifying Trigonometric Expressions

We use a combination of trigonometric identities, formulas, and techniques from algebra to manipulate and simplify trigonometric expressions.

Example 3.1.2.

Simplify
\begin{equation*} \tan^2(x)\cdot\csc^2(x) \end{equation*}

Solution.

We can simplify this expression by writing each function in terms of sine and cosine functions
\begin{equation*} \tan^2(x)\cdot\csc^2(x)=\frac{\sin^2(x)}{\cos^2(x)}\cdot\frac{1}{\sin^2(x)}=\frac{1}{\cos^2(x)}=\sec^2(x) \end{equation*}

Example 3.1.3.

Simplify
\begin{equation*} \sin^2(x)(\cot^2(x) - 1) \end{equation*}

Solution.

We can simplify this expression by first using the Pythagorean Identity and then using the Reciprocal Identity:
\begin{equation*} \sin^2(x)(\cot^2(x) - 1) = \sin^2(x)(-\csc^2(x)) = \sin^2(x)\left(-\frac{1}{\sin^2(x)}\right) = -1 \end{equation*}

Subsection 3.1.3 Verifying Trigonometric Identities

To verify trigonometric identities, we begin with an expression on one side of the equation and manipulate it using trigonometric identities and algebraic techniques until it matches the expression on the other side.

Remark 3.1.4. Steps for Verifying Trigonometric Identities.

To verify that an equation is an identity:
  1. Pick an expression on one side of the equation. Often it is the more complicated expression.
  2. Transform the expression using techniques such as trigonometric identities, rewriting in terms of sine and cosine functions, factoring, common denominator, or multiplying numerator and denominator by the same term.
  3. Continue manipulating until the transformed expression matches the other side of the equation.
  4. If you have trouble making one side resemble the other, try manipulating both sides separately, and make them match to reach the same result.
Note: Unlike solving equations where we perform the same operation on both sides of the equal sign, when verifying trigonometric identities, we work with only one side and manipulate it to resemble the other side.

Example 3.1.5. Verify the Identity by Rewriting in Terms of Sine and Cosine.

Verify the identity
\begin{equation*} \frac{\sin(x)}{\tan(x)}=\cos(x) \end{equation*}

Solution.

We use the Quotient Identity to rewrite \(\tan(x)\) in terms of \(\sin(x)\) and \(\cos(x)\text{:}\)
\begin{equation*} \frac{\sin(x)}{\tan(x)}=\dfrac{\sin(x)}{\frac{\sin(x)}{\cos(x)}}=\cancel{\sin(x)}\cdot\frac{\cos(x)}{\cancel{\sin(x)}}=\cos(x)\text{.} \end{equation*}

Example 3.1.6. Verify the Identity by Factoring.

Verify the identity
\begin{equation*} \cos^4(x)+\sin^2(x)\cos^2(x)=\cos^2(x)\text{.} \end{equation*}

Solution.

First notice that both terms in \(\cos^4(x)+\sin^2(x)\cos^2(x)\) contain \(\cos^2(x)\text{.}\) Then
\begin{align*} \cos^4(x)+\sin^2(x)\cos^2(x) \amp = \cos^2(x)\cdot\cos^2(x)+\sin^2(x)\cos^2(x) \\ \amp = \cos^2(x)\cdot\left(\cos^2(x)+\sin^2(x)\right) \\ \amp = \cos^2(x)\cdot1 \\ \amp = \cos^2(x) \end{align*}

Example 3.1.7. Verify the Identity by Odd-Even Properties.

Verify the identity
\begin{equation*} \frac{\cos(x)-\sin(x)}{\cos(-x)+\sin(-x)}=1 \end{equation*}

Solution.

By the Odd-Even Properties, we have \(\sin(-x)=-\sin(x)\) and \(\cos(-x)=\cos(x)\text{.}\) Thus,
\begin{equation*} \frac{\cos(x)-\sin(x)}{\cos(-x)+\sin(-x)}=\frac{\cos(x)-\sin(x)}{\cos(x)-\sin(x)}=1 \end{equation*}

Example 3.1.8. Verify the Identity by Multiplying the Numerator and Denominator by the Same Term.

Verify the identity
\begin{equation*} \frac{\sin(x)}{\sin(x)+\cos(x)}=\frac{1}{1+\cot(x)} \end{equation*}

Solution.

Multiplying both the numerator and denominator by \(\frac{1}{\sin(x)}\text{,}\) we get
\begin{align*} \frac{\sin(x)}{\sin(x)+\cos(x)}\cdot\frac{\frac{1}{\sin(x)}}{\frac{1}{\sin(x)}} \amp = \frac{\cancel{\sin(x)}\cdot\frac{1}{\cancel{\sin(x)}}}{\cancel{\sin(x)}\cdot\frac{1}{\cancel{\sin(x)}}+\cos(x)\cdot\frac{1}{\sin(x)}} \\ \amp = \frac{1}{1+\frac{\cos(x)}{\sin(x)}} \\ \amp =\frac{1}{1+\cot(x)} \end{align*}

Example 3.1.9. Verify the Identity by Manipulating Both Sides Separately.

Verify the identity
\begin{equation*} \frac{1-\cos x}{1+\cos x}=(\csc x-\cot x)^2\text{.} \end{equation*}

Solution.

We begin by simplifying the right-hand side of the equation
\begin{align*} (\csc x-\cot x)^2 \amp = \csc^2x-2\csc x\cot x+\cot^2x \\ \amp = \csc^2x+\cot^2x-2\csc x\cot x \\ \amp = \csc^2x+\cot^2x-2\frac{1}{\sin x}\frac{\cos x}{\sin x} \\ \amp = \csc^2x+\cot^2x-2\frac{\cos x}{\sin^2 x} \text{.} \end{align*}
Next, we will manipulate the left-hand side of the equation to get it to look like \(\csc^2x+\cot^2x-2\frac{\cos x}{\sin^2 x}\text{.}\)
\begin{align*} \frac{1-\cos x}{1+\cos x} \amp = \frac{(1-\cos x)(1-\cos x)}{(1+\cos x)(1-\cos x)} \\ \amp = \frac{1-2\cos x+\cos^2x}{1-\cos^2x} \\ \amp = \frac{1+\cos^2x-2\cos x}{\sin^2x} \\ \amp = \frac{1}{\sin^2x}+\frac{\cos^2x}{\sin^2x}-2\frac{\cos x}{\sin^2x} \\ \amp = \csc^2x+\cot^2x-2\frac{\cos x}{\sin^2 x} \text{.} \end{align*}
Thus, since the left-hand side and the right-hand side of the equation can both be manipulated to \(\csc^2x+\cot^2x-2\frac{\cos x}{\sin^2 x}\text{,}\) we have established the identity.

Exercises 3.1.4 Exercises

Exercise Group.

Verify the identity.
1.
\(\cos\theta\sec\theta=1\)
2.
\(\cos x\csc x=\cot x\)
3.
\(\frac{\cos\theta\sec\theta}{\tan\theta}=\cot\theta\)
4.
\(\frac{\cot t\tan t}{\csc t}=\sin t\)
5.
\((1+\tan\theta)(1-\tan\theta)+\sec^2\theta=2\)
6.
\(1- \sin^2(x) =\cos^2(x)\)
7.
\(1 - \sec^2(\theta)=-\tan^2(\theta)\)
8.
\(\tan(t) \cdot \cot(t)=1\)
9.
\((\sin\theta+\cos\theta)^2=1+2\sin\theta\cos\theta\)
10.
\((1-\cot\theta)^2=\csc^2\theta-2\cot\theta\)
11.
\(\frac{\sin\theta}{\csc\theta}+\frac{\cos\theta}{\sec\theta}=1\)
12.
\(\sin^2 t(\csc^2 t+\sec^2 t)=\sec^2 t\)
13.
\(\sin^2(x) - \sin^2(x) \cos^2(x)=\sin^4(x)\)
14.
\(\sin^2(-x)+\cos^2(-x)=1\)
15.
\(\cos(-t)+\sin(-t)=\cos(t)-\sin(t)\)
16.
\((\sin\theta+\cos\theta)^2-2\sin\theta\cos\theta=1\)
17.
\(\cot^2 x(\sec^2 x-1)=1\)
18.
\((1+\sin(t))(1+\sin(-t))=\cos^2 t\)
19.
\(\tan^4\theta=\tan^2\theta\sec^2\theta-\tan^2\theta\)
20.
\(\frac{1}{1-\sin x}+\frac{1}{1+\sin x}=2\sec^2x\)
21.
\(\frac{1}{\csc t+1}-\frac{1}{\csc t-1}=-2\tan^2t\)
22.
\(\frac{1}{1-\cos\theta}+\frac{1}{1+\cos\theta}=2\csc^2\theta\)
23.
\(\frac{1}{1-\cos\theta}+\frac{1}{1+\cos\theta}=2+2\cot^2\theta\)
24.
\(\frac{1-\cos^2 x}{\cos x}=\sin x\tan x\)
25.
\(1-\frac{\cos^2\theta}{1+\sin\theta}=\sin\theta\)
26.
\(\sec^2 t+\csc^2 t=\csc^2t\sec^2t\)
27.
\(\frac{1+\tan x}{1-\tan x}=\frac{\cot x+1}{\cot x-1}\)
28.
\(\frac{\cos\theta}{1-\sin\theta}=\frac{1+\sin\theta}{\cos\theta}\)
29.
\(1-\frac{\cos^2 t}{1+\sin t}=\sin t\)
30.
\(\frac{1+\cos\theta}{\cos\theta}=\frac{\tan^2\theta}{\sec\theta-1}\)
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