Skip to main content
Contents Dark Mode Prev Up Next \(\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}
\newcommand{\set}[1]{\left\{ #1 \right\}}
\renewcommand{\neg}{\sim}
\newcommand{\brac}[1]{\left( #1 \right)}
\newcommand{\rad}[1]{#1 \ \text{rad}}
\newcommand{\eval}[1]{\left. #1 \right|}
\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}
\newcommand{\ang}[1]{#1^\circ}
\newcommand{\crossmethod}[4]{
\begin{tikzpicture}[baseline=(M.base)]
\node (M) at (0,0) {$#1$};
\node (P) at (0,-1) {$#2$};
\node (N) at (1.5,0) {$#3$};
\node (Q) at (1.5,-1) {$#4$};
\draw (M) -- (Q);
\draw (P) -- (N);
\end{tikzpicture}
}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 15.2 Intro to Proving Trigonometric Identities
We already know a few fundamental trig identities. Using these, we can
prove new and more complicated identities.
Proving a trigonometric identity involves manipulating each side of the equation algebraically, to show that both sides of the equation are equivalent.
Subsection 15.2.1 General Principles for Proving Identities
This topic is often considered one of the most challenging by students, because:
It involves a culmination of a variety of algebra techniques learned over the course of high school math.
It requires some creativity to choose a strategy.
It involves more abstract and symbolic reasoning, with basically no visualizations.
Here are some general principles:
Rewrite everything in terms of sine and cosine . All trigonometric ratios can be written in terms of the “basic” trig ratios sine and cosine.
The one exception is if the entire identity includes
only tangent and/or cotangent, in which case it is usually simpler to convert everything in terms of tangent.
Start with the more complicated side . In general,
Tangent is more complicated than sine and cosine.
Reciprocal trig functions are more complicated than primary trig functions (sine, cosine, tangent).
Multiple terms are more complicated than a single term.
You want to start with the more complicated side, because it is easier to start from something complex and simplify it, rather than start from something simple and make it more complex.
Begin with the end in mind . Look at the form of the desired side before you start, and guide your work toward that structure. Especially helpful if you are stuck and unsure how to continue.
Here are some particular algebraic tactics:
Combine fractions, using a common denominator.
Simplify complex fractions, by clearing denominators (multiply the numerator and denominator by the LCD).
Use algebraic identities, e.g. difference of squares, square of a sum, etc.
\begin{equation*}
a^2 - b^2 = (a - b)(a + b)
\end{equation*}
\(1 - \sin^2{x} = (1 - \sin{x})(1 + \sin{x})\text{.}\)
\(\sin^2{x} - \cos^2{x} = (\sin{x} - \cos{x})(\sin{x} + \cos{x})\text{.}\)
Convert
\(\sin^2{x}\) to
\(\cos^2{x}\) or vice versa, using the Pythagorean identity:
\begin{align*}
\sin^2{x} \amp = 1 - \cos^2{x}\\
\cos^2{x} \amp = 1 - \sin^2{x}
\end{align*}
Multiply by the conjugate, in order to use the Pythagorean identity (converting
\(1-\cos^2{x}\) to
\(\sin^2{x}\text{,}\) or
\(1-\sin^2{x}\) to
\(\cos^2{x}\) ). In particular, if the identity includes,
\begin{equation*}
1 - \sin{x}, \quad 1 + \sin{x}, \quad 1 - \cos{x}, \quad \text{or} \quad 1 + \cos{x}
\end{equation*}
Use all trigonometric identities available to you. When practicing, have a list of them for reference, and you’ll naturally be more comfortable with them over time.
Subsection 15.2.2 Examples
Exercise Group 15.2.1 . Basic Examples.
Prove each trigonometric identity.
(a) \(\sin{x}\cot{x}=\cos{x}\) Hint .
Rewrite in terms of sine and cosine, cancel common factors.
(b) \(\cos{x} \csc{x} \tan{x} = 1\) Hint .
Rewrite in terms of sine and cosine, cancel common factors.
(c) \(\dfrac{\sec{x}}{\csc{x}} + \dfrac{\sin{x}}{\cos{x}} = 2\tan{x}\) Hint .
Rewrite in terms of sine and cosine, simplify fractions.
(d) \(\cos{x}\tan{x}=\sin{x}\) Hint .
Rewrite in terms of sine and cosine, cancel common factors.
(e) \(\cot{x} \sin{x} \sec{x} = 1\) Hint .
Rewrite in terms of sine and cosine, cancel common factors.
(f) \(\cos{x}\brac{\csc{x}-\sec{x}}=\cot{x}-1\) Hint .
Expand, rewrite in terms of sine and cosine.
(g) \(\sin{x} \cos{x} \tan{x} = \dfrac{1}{\csc^2{x}}\) Hint .
Work with both sides, rewrite in terms of sine and cosine, cancel common factors.
Exercise Group 15.2.2 . Practice with the Pythagorean Identity.
Prove each trigonometric identity.
(a) \(\csc^{2}x(1-\cos^{2}x)=1\) Hint .
Pythagorean identity, cancel common factors.
(b) \(\sin^{2}{x}\brac{1 + \cot^{2}{x}} = 1\) Hint .
Pythagorean identity, rewrite in terms of sine and cosine.
(c) \(\dfrac{\sin^{2}{x}+\cos^{2}{x}}{\cos{x}}=\sec{x}\) Hint .
Pythagorean identity, rewrite in terms of sine and cosine.
(d) \((\tan x-1)^{2}=\sec^{2}x-2\tan x\) Hint .
Expand, Pythagorean identity.
(e) \(\sin{x}\tan{x} + \cos{x} = \sec{x}\) Hint .
Start with LHS, combine fractions, Pythagorean identity.
(f) \(\csc{x} = \cos{x}\cot{x} + \sin{x}\) Hint .
Combine fractions, Pythagorean identity.
(g) \(\sin{x} \cos{x} \cot{x} = 1 - \sin^2{x}\) Hint .
Rewrite in terms of sine and cosine, cancel common factors, Pythagorean identity.
(h) \(\cos{x}(\csc{x}+\sec{x})=\cot{x}+1\) Hint .
Rewrite in terms of sine and cosine, distribute.
(i) \(\dfrac{\sin{x}-1}{\cos{x}}=\tan{x}-\sec{x}\) Hint .
Start with either side, combine fractions (or split fraction).
(j) \(1-\sin{x}\cos{x}\tan{x}=\cos^{2}{x}\) Hint .
Rewrite in terms of sine and cosine, cancel common factors, Pythagorean identity.
(k) \(\sin{x}+\cos{x}\cot{x}=\csc{x}\) Hint .
Rewrite in terms of sine and cosine, combine fractions with a common denominator, Pythagorean identity.
(l) \(\sin^2{x}+\sin^2{x}\cot^2{x}=1\) Hint .
Factor out common factor, Pythagorean identity (or rewrite in terms of sine and cosine, cancel common factors, Pythagorean identity).
(m) \(\dfrac{\cos{x}}{\sec{x}} + \dfrac{\sin{x}}{\csc{x}} = 1\) Hint .
Rewrite in terms of sine and cosine, Pythagorean identity.
(n) \(\dfrac{1}{\sec{x}\tan{x}} = \csc{x} - \sin{x}\) Hint .
Work with both sides, Pythagorean identity, combine fraction with a common denominator.
(o) \(\sec{x} - \cos{x} = \dfrac{\sin^2{x}}{\cos{x}}\) Hint .
Rewrite in terms of sine and cosine, combine fractions, Pythagorean identity.
Exercise Group 15.2.3 . Mixed Identity Proofs I.
Prove each trigonometric identity.
(a) \(\dfrac{\sin^2{x}}{1 - \cos{x}} = 1 + \cos{x}\) Hint .
Pythagorean identity, factor difference of squares, cancel common factors.
(b) \(\dfrac{1}{\csc{x}} = \sin{x} \cos^2{x} + \sin^3{x}\) Hint .
Factor common factor, rewrite in terms of sine and cosine, Pythagorean identity.
(c) \(\dfrac{1+\sin{x}}{\sin{x}}=1+\csc{x}\) Hint .
Start with either side, combine fractions with a common denominator (or split fraction).
(d) \(\dfrac{\cos{x}}{1 + \sin{x}} = \dfrac{1 - \sin{x}}{\cos{x}}\) Hint .
Multiply by the conjugate, Pythagorean identity, cancel common factors.
(e) \(\dfrac{\sec{x} - \csc{x}}{\sec{x}\csc{x}} = \sin{x} - \cos{x}\) Hint .
Rewrite in terms of sine and cosine, simplify complex fractions by clearing denominators, simplify and cancel common factors.
(f) \(\dfrac{\sin^{2}x+\cos^{2}x}{\sec x}=\cos x\) Hint .
Pythagorean identity, rewrite in terms of sine and cosine, simplify.
(g) \(\dfrac{1}{1-\sin{x}}=\dfrac{1+\sin{x}}{\cos^2{x}}\) Hint .
Start with either side, multiply by the conjugate, Pythagorean identity, cancel common factors.
(h) \(\dfrac{\cos x+1}{\sin x+\tan x}=\cot x\) Hint .
Rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, factor common factors, cancel common factors.
(i) \(\dfrac{1+\sin x}{\cos x} = \dfrac{\cos x}{1-\sin x}\) Hint .
Multiply by the conjugate, Pythagorean identity, cancel common factors.
(j) \(\dfrac{1+\cos x}{\sin x}=\dfrac{\sin x}{1-\cos x}\) Hint .
Multiply by the conjugate, Pythagorean identity, cancel common factors.
(k) \(\cos x+\cos x \tan^{2}x=\sec x\) Hint .
Factor common factors, Pythagorean identity.
(l) \(1-2\sin^{2}{x}=2\cos^{2}{x}-1\) Hint .
Pythagorean identity, simplify.
(m) \(\dfrac{\sin{x}+\tan{x}}{1+\cos{x}}=\tan{x}\) Hint .
Rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, factor out common factors, cancel common factors.
Exercise Group 15.2.4 . Mixed Identity Proofs II.
Prove each trigonometric identity.
(a) \(\dfrac{\cos{x} + \cot{x}}{1 + \sin{x}} = \cos{x} \csc{x}\) Hint .
Rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, factor and cancel common factors.
(b) \(\dfrac{1 + \sin{x}}{1 + \csc{x}} = \sin{x}\) Hint .
Simplify complex fraction by clearing denominators, factor and cancel common factors.
(c) \(\dfrac{\cos{x}}{1-\sin{x}} = \tan{x} + \sec{x}\) Hint .
Can start with either side, multiply by the conjugate, Pythagorean identity.
(d) \(\tan{x} + \cot{x} = \sec{x}\csc{x}\) Hint .
Combine fractions with a common denominator, Pythagorean identity.
(e) \(\dfrac{1 - 2\cos^{2}{x}}{\sin{x}\cos{x}} = \tan{x} - \cot{x}\) Hint .
Start with the RHS, rewrite in terms of sine and cosine, combine fractions with a common denominator, Pythagorean identity.
(f) \(\dfrac{1+\tan x}{1+\cot x}=\tan x\) Hint .
Rewrite in terms of tangent, simplify complex fraction by clearing denominators, cancel common factors.
(g) \(\dfrac{\sec x}{\sin x}-\dfrac{\sin x}{\cos x}=\cot x\) Hint .
Rewrite in terms of sine and cosine, combine fractions with a common denominator, Pythagorean identity, simplify.
(h) \(\dfrac{\cot x+\tan x}{\sec x}=\csc x\) Hint .
Rewrite in terms of sine and cosine, combine fractions with a common denominator, cancel common factors.
(i) \(\dfrac{1-\cos{x}}{\sin{x}}=\dfrac{\tan{x}-\sin{x}}{\tan{x}\sin{x}}\) Hint .
Start with the RHS, rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, cancel common factors.
(j) \(\dfrac{2}{1-\sin{x}}+\dfrac{2}{1+\sin{x}}=4\sec^2{x}\) Hint .
Combine fractions with a common denominator, combine like terms, Pythagorean identity.
(k) \(\dfrac{\sec{x} - \cos{x}}{\tan{x}} = \sin{x}\) Hint .
Rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, combine fractions with a common denominator, Pythagorean identity.
(l) \(\sec^{4}x-\sec^{2}x=\tan^{4}x+\tan^{2}x\) Hint .
Factor common factors, Pythagorean identity.
(m) \(\brac{1+\csc{x}}\brac{1-\sin{x}}=\cot{x}\cos{x}\) Hint .
Expand, rewrite in terms of sine and cosine, cancel common factors, combine fractions with a common denominator, Pythagorean identity.
(n) \(\dfrac{\tan{x} \cos^2{x}}{\sec{x}} = \sin{x} - \sin^3{x}\) Hint .
Start with the LHS, rewrite in terms of sine and cosine, Pythagorean identity.
(o) \(\dfrac{1}{1+\sin{x}} + \dfrac{1}{1-\sin{x}} = 2\sec^2{x}\) Hint .
Start with the LHS, combine fraction with a common denominator.
(p) \(\dfrac{\sin{x} - \csc{x}}{1 + \sin{x}} = \dfrac{\sin{x} - 1}{\sin{x}}\) Hint .
Rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, factor and cancel common factors.
(q) \(\dfrac{\sin{x} \cos{x}}{1 + \cos{x}} = \dfrac{1-\cos{x}}{\tan{x}}\) Hint .
Multiply by the conjugate, Pythagorean identity, cancel common factors.
Exercise Group 15.2.5 . Advanced Identity Proofs I.
Prove each trigonometric identity.
(a) \(\dfrac{\tan{x}}{\tan{x}+\sin{x}}=\dfrac{1-\cos{x}}{\sin^2{x}}\) Hint .
Work with both sides, rewrite in terms of sine and cosine, simplify complex fractions by clearing denominators, cancel common factors, factor difference of squares (or multiply by the conjugate).
(b) \(\cos^2{x}-\sin^2{x}=\dfrac{\cot{x}-\tan{x}}{\cot{x}+\tan{x}}\) Hint .
Rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, Pythagorean identity.
(c) \(\csc{x}-\dfrac{\sin{x}}{1+\cos{x}}=\cot{x}\) Hint .
Rewrite in terms of sine and cosine, combine fractions, Pythagorean identity, factor out common factor, cancel common factors.
(d) \(\sin^4{x}-\cos^4{x}=1-2\cos^2{x}\) Hint .
Factor difference of squares, Pythagorean identity.
(e) \(\cot{x}-\tan{x}=\dfrac{2\cos^2{x}-1}{\sin{x}\cos{x}}\) Hint .
Start with either side, rewrite in terms of sine and cosine, combine fractions with a common denominator.
(f) \(\dfrac{1-\sin{x}}{1+\sin{x}}=(\sec{x}-\tan{x})^2\) Hint .
Start with the RHS, rewrite in terms of sine and cosine, combine fractions with a common denominator, Pythagorean identity.
(g) \(\tan{x} + \dfrac{\cos{x}}{1 + \sin{x}} = \sec{x}\) Hint .
Combine fractions with a common denominator, Pythagorean identity, cancel common factors.
(h) \(\dfrac{\sin{x}+\cos{x}\cot{x}}{\cos{x}\csc{x}}=\sec{x}\) Hint .
Rewrite in terms of sine and cosine, simplify complex fractions by clearing denominators, Pythagorean identity.
(i) \(\csc^2{x} + \sec^2{x} = \csc^2{x} \sec^2{x}\) Hint .
Rewrite in terms of sine and cosine, combine fractions, Pythagorean identity.
(j) \(\sin^{4}{x}-\cos^{4}{x}=2\sin^{2}{x}-1\) Hint .
Difference of squares, Pythagorean identity.
(k) \(\dfrac{\sec^4{x}-1}{\tan^2{x}}=2+\tan^2{x}\) Hint .
Difference of squares, Pythagorean identity, cancel common factors.
(l) \(\dfrac{\sin{x}+\cos{x}}{\csc{x}+\sec{x}}=\sin{x}\cos{x}\) Hint .
Rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, cancel common factors.
(m) \(\dfrac{\cos{x}+\sin{x}}{\cos{x}-\sin{x}}=\dfrac{1+\tan{x}}{1-\tan{x}}\) Hint .
Rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators.
(n) \(\dfrac{\csc{x}+\cot{x}}{\tan{x}+\sin{x}}=\cot{x}\csc{x}\) Hint .
Start with the LHS, rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, cancel common factors.
Exercise Group 15.2.6 . Advanced Identity Proofs II.
Prove each trigonometric identity.
(a) \(\dfrac{1 + \sin{x}}{\cos{x}} + \dfrac{\cos{x}}{1 + \sin{x}} = \dfrac{2}{\cos{x}}\) Hint .
Combine fractions with a common denominator, Pythagorean identity, cancel common factors.
(b) \(\dfrac{1+\sec{x}}{\sin{x}+\tan{x}}=\csc{x}\) Hint .
Rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, cancel common factors.
(c) \(\dfrac{\sec{x}}{1-\sin{x}}=\dfrac{1+\sin{x}}{\cos^3{x}}\) Hint .
Start with the LHS, rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, multiply by the conjugate, Pythagorean identity.
(d) \(\cos^2{x}=\dfrac{1-2\sin^2{x}}{1-\tan^2{x}}\) Hint .
Rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, Pythagorean identity, cancel common factors.
(e) \(\dfrac{1-\cos{x}}{\sin{x}}=\dfrac{1}{\csc{x}+\cot{x}}\) Hint .
Start with the RHS, rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, multiply by the conjugate, Pythagorean identity.
(f) \(\dfrac{\sin^2{x}-\tan{x}}{\cos^2{x}-\cot{x}}=\tan^2{x}\) Hint .
Rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, factor and cancel common factors.
(g) \(\dfrac{\tan{x} - \cot{x}}{\tan{x} + \cot{x}} + 1 = 2\sin^{2}{x}\) Hint .
Simplify complex fraction by clearing denominators, Pythagorean identity.
(h) \(\dfrac{1+\cos{x}}{\tan{x} + \sec{x}} = \cot{x}\) Hint .
Rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, factor out common factor, cancel common factors.
(i) \(\dfrac{\tan{x}-\sin{x}}{\sin^3{x}} = \dfrac{\sec{x}}{1+\cos{x}}\) Hint .
Start with the LHS but RHS works also, simplify complex fraction by clearing denominators, cancel common factors, Pythagorean identity, factor and cancel.
(j) \(\dfrac{\cos{x}}{\sec{x}-1}+\dfrac{\cos{x}}{\sec{x}+1}=2\cot^{2}x\) Hint .
Rewrite in terms of sine and cosine, common denominator, Pythagorean identity, simplify.
(k) \(\dfrac{\tan{x}}{\cos{x} - \sec{x}} = -\csc{x}\) Hint .
Rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, Pythagorean identity, simplify and cancel.
(l) \(\csc{x}\brac{\csc{x}+\cot{x}}=\dfrac{1}{1-\cos{x}}\) Hint .
Expand, rewrite in terms of sine and cosine, combine fractions, Pythagorean identity, factor difference of squares (or multiply by the conjugate), cancel common factors.
(m) \(\tan^{4}{x}-\sec^{4}{x}=1-2\sec^{2}{x}\) Hint .
Difference of squares, Pythagorean identity, simplify.
(n) \(\dfrac{\sec{x}-1}{\sin^2{x}}=\dfrac{\sec^2{x}}{1+\sec{x}}\) Hint .
Work with both sides, rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, factor difference of squares (or multiply by the conjugate), cancel.
(o) \(\dfrac{\cos^4{x}-\sin^4{x}}{1-\tan^4{x}}=\cos^4{x}\) Hint .
Rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, cancel common factors.
(p) \(\dfrac{\sec{x}}{1-\cos{x}}=\dfrac{\sec{x}+1}{\sin^2{x}}\) Hint .
Start with either side, rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, multiply by the conjugate (or factor difference of squares), Pythagorean identity.
(q) \(\dfrac{\cos{x}}{\csc{x}+1}+\dfrac{\cos{x}}{\csc{x}-1}=2\tan{x}\) Hint .
Rewrite in terms of sine and cosine, simplify complex fraction by clearing denominators, combine fraction with a common denominator.
(r) \(\tan{x}(\csc{x}+1)=\dfrac{\cot{x}}{\csc{x}-1}\) Hint .
Work with both sides, rewrite in terms of sine and cosine, distribute, combine fractions with a common denominator, simplify complex fraction by clearing denominators, multiply by the conjugate, Pythagorean identity.
(s) \(\dfrac{\cos{x}-\cos{y}}{\sin{x}+\sin{y}}+\dfrac{\sin{x}-\sin{y}}{\cos{x}+\cos{y}}=0\) Hint .
Combine fractions with a common denominator, Pythagorean identity.
The one exception is if the entire identity includes only tangent and/or cotangent, in which case it is usually simpler to convert everything in terms of tangent.
