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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 all valid values of its variables. We begin by reviewing some of the basic trigonometric identities from
ChapterΒ 1 , collectively known as the fundamental trigonometric identities.
Definition 3.1.1 .
The
fundamental trigonometric identities are:
\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*}
\begin{align*}
\tan\theta \amp =\dfrac{\sin\theta}{\cos\theta} \amp \cot\theta\amp =\dfrac{\cos\theta}{\sin\theta}
\end{align*}
\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*}
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*}
\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)\text{.}
\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)\text{.}
\end{equation*}
Example 3.1.3 .
Simplify
\begin{equation*}
\sin^2(x)(\cot^2(x) - 1)\text{.}
\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\text{.}
\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.
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)\text{.}
\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) \text{.}
\end{align*}
Example 3.1.7 . Verify the Identity by Even-Odd Properties.
Verify the identity
\begin{equation*}
\frac{\cos(x)-\sin(x)}{\cos(-x)+\sin(-x)}=1
\end{equation*}
Solution .
By the Even-Odd 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\text{.}
\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)}\text{.}
\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)} \text{.}
\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 simplify it into \(\csc^2x+\cot^2x-2\frac{\cos x}{\sin^2 x}\text{.}\)
\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 equal
\(\csc^2x+\cot^2x-2\frac{\cos x}{\sin^2 x}\text{,}\) we have established the identity.
Exercises 3.1.4 Exercises
Exercise Group.
1.
\(\cos\theta\sec\theta=1\)
2. 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.
\(\frac{\tan^2 t}{\sec t+1}=\frac{1-\cos t}{\cos t}\)
30.
\(\frac{1+\cos\theta}{\cos\theta}=\frac{\tan^2\theta}{\sec\theta-1}\)
