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Section 15.1 Intro to Trigonometric Identities
In algebra, there are various equations which are always true, called identities.
Example 15.1.1 .
\begin{equation*}
(a + b)^2 = a^2 + 2ab + b^2
\end{equation*}
These two sides are exactly the same, for any value of
\(a\) and
\(b\text{.}\) In other words, it is true no matter what numbers
\(a\) and
\(b\) are. This is called an identity. Another identity is,
\begin{equation*}
(a-b)(a+b) = a^2 - b^2
\end{equation*}
We can also consider
trigonometric identities , which are trigonometric equations that are true for all values of the variable.
We already know some basic identities, which could almost be called definitions.
\begin{align*}
\amp \boxed{\begin{gathered}\textbf{Reciprocal identities}\\[2pt]\csc\theta=\dfrac{1}{\sin\theta}\qquad\sec\theta=\dfrac{1}{\cos\theta}\qquad\cot\theta=\dfrac{1}{\tan\theta}\end{gathered}}\\
\\
\amp \boxed{\begin{gathered}\textbf{Quotient identities}\\[2pt]\tan\theta=\dfrac{\sin\theta}{\cos\theta}\qquad\cot\theta=\dfrac{\cos\theta}{\sin\theta}\end{gathered}}
\end{align*}
Subsection 15.1.1 Pythagorean Identities
There is also the Pythagorean identity, which comes from the unit circle,
\begin{equation*}
\boxed{\sin^2{x} + \cos^2{x} = 1} \qquad \text{for all $x$}
\end{equation*}
This basically means that if you see
\(\sin^2{x} + \cos^2{x}\) in an expression, you can replace it with simply 1.
This identity can also be rearranged, to isolate for either
\(\sin^2{x}\) or
\(\cos^2{x}\text{,}\)
\begin{align*}
\sin^2{x} \amp = 1 - \cos^2{x}\\
\cos^2{x} \amp = 1 - \sin^2{x}
\end{align*}
We can also derive further identities, using this identity,
\begin{equation*}
\boxed{1 + \tan^2{x} = \sec^2{x}} \qquad \text{and} \qquad \boxed{1 + \cot^2{x} = \csc^2{x}}
\end{equation*}
At first, when you’re solving problems, have all of these identities available as a reference. Here is a summary of all common trigonometric identities:
Summary of Trigonometric Identities . We will learn the remaining identities in the next few sections. On a test, most instructors will give you a formula sheet with all of the trigonometric identities (similar to my summary). However, with practice, you will naturally remember most of them.
Subsection 15.1.2 Simplifying Trigonometric Expressions
Exercise Group 15.1.1 . Simplifying Expressions I.
Simplify each trigonometric expression.
(a) \(\sec{x} \sin{x}\)
Hint .
Convert everything to sine and cosine, cancel common factors.
Answer .
(b) \(\cot{x}\sin{x}\)
(c) \((1 - \tan{x})(1 + \cot{x})\)
(d) \(\dfrac{\sec{x}}{\tan{x}}\)
Hint .
Convert everything to sine and cosine, cancel common factors.
Answer .
(e) \(\sin{x} \cos{x} (\csc{x} + \sec{x})\)
Hint .
Distribute, convert everything to sine and cosine, cancel common factors.
Answer .
(f) \(\sin{x} \cos{x} \sec{x} \csc{x}\)
Hint .
Convert everything to sine and cosine, cancel common factors.
Answer .
Exercise Group 15.1.2 . Simplifying Expressions II.
Simplify each trigonometric expression.
(a) \((\tan{x} + \sec{x})(\tan{x} - \sec{x})\)
Hint .
Expand, Pythagorean identity.
Answer .
(b) \(\cos^4{x} - \sin^4{x}\)
Hint .
Factor difference of squares, Pythagorean identity.
Answer .
\(\cos^2{x} - \sin^2{x}\)
(c) \(\dfrac{\cot^2{x}}{1 - \sin^2{x}}\)
Hint .
Convert everything to sine and cosine, Pythagorean identity.
Answer .
(d) \((\tan{x} + \cot{x})^2 - (\tan{x} - \cot{x})^2\)
Hint .
Ideally factor difference of squares, if not then expand and collect like terms.
Answer .
(e) \(\dfrac{1}{\cos{x} + 1} - \dfrac{1}{\cos{x} - 1}\)
Hint .
Combine fractions with a common denominator, collect like terms, Pythagorean identity.
Answer .
(f) \(\dfrac{\csc{x} - \sin{x}}{\cot{x}}\)
Hint .
Convert everything to sine and cosine, combine fractions, Pythagorean identity, cancel common factors.
Answer .
(g) \(\dfrac{\tan{x} - \tan{x}\sin^2{x}}{\cos^2{x}}\)
Hint .
Factor common factors, Pythagorean identity, cancel common factors.
Answer .
(h) \(\sin{x}\cot{x}-\sin{x}\cos{x}\)
(i) \(\dfrac{\sec^2{x} - \tan^2{x}}{\sin^2{x} + \cos^2{x}}\)
(j) \(\dfrac{\sec{x} \tan{x}}{\csc{x} \tan{x} \sec{x} - 1}\)
Hint .
Convert everything to sine and cosine, simplify complex fraction by clearing denominators, Pythagorean identity.
Answer .
(k) \(\dfrac{1 + \tan^2{x}}{1 + \cot^2{x}}\)
Hint .
Convert everything to tangent, simplify complex fraction by clearing denominators, cancel common factors.
Answer .
(l) \(\dfrac{\sin^2{x} + \cos^2{x}}{\sec{x}}\)
Hint .
Pythagorean identity, convert everything to sine and cosine.
Answer .
(m) \(\dfrac{\sec^2{x} - \tan^2{x}}{\tan{x}}\)
Hint .
Pythagorean identity, convert everything to sine and cosine.
Answer .
(n) \(\dfrac{\tan{x} \csc^2{x} - \tan{x} \cot^2{x}}{\sin^2{x} \sec{x} \csc{x} \cot{x}}\)
Hint .
Factor common factors, convert everything to sine and cosine, cancel common factors, Pythagorean identity.
Answer .
(o) \(\dfrac{\cos{x}}{\cos{x} - \cos^3{x}}\)
Hint .
Factor common factors, Pythagorean identity.
Answer .
(p) \(\dfrac{\tan{x} \csc^2{x}}{\sec^2{x}}\)
Hint .
Convert everything to sine and cosine, cancel common factors.
Answer .
(q) \(\sec{x} \cot{x} \sin^2{x}\)
Hint .
Convert everything to sine and cosine, cancel common factors.
Answer .
(r) \((\sin{x} + \cos{x})^2 + (\sin{x} - \cos{x})^2\)
Hint .
Expand, cancel like terms, Pythagorean identity.
Answer .
Exercise Group 15.1.3 . Write as a Single Trigonometric Ratio.
Simplify each expression as a single trigonometric ratio.
(a) \(\dfrac{\sin^2{x}}{\cos^2{x}} + 1\)
(b) \((1-\sin{x})(1+\sin{x})\)
(c) \(\dfrac{\tan{x}}{\sin{x}}\)
(d) \(\sin{x} + \cot{x} \cos{x}\)
(e) \(\dfrac{\cos{x}}{1-\sin^2{x}}\)
(f) \(\dfrac{\tan^2{x} + 1}{\sec{x}}\)
(g) \(\cos{x}(1+\sec{x})(\cos{x}-1)\)
Exercise Group 15.1.4 . Factor Each Trigonometric Expression.
Factor each trigonometric expression, and simplify if possible.
(a) \(1-\cos^2{x}\) Answer .
\((1-\cos{x})(1+\cos{x})\)
(b) \(\sin^2{x}-2\sin{x}+1\)
(c) \(\cos{x}-\cos^2{x}\)
(d) \(\sin^2{x} - \cos^2{x}\)
Hint . Answer .
\((\sin{x} - \cos{x})(\sin{x} + \cos{x})\)
(e) \(\dfrac{\csc^2{x}-3\csc{x}+2}{\csc^2{x}-1}\) Answer .
\(\dfrac{\csc{x}-2}{\csc{x}+1}\)
(f) \(\cos^3{x} - \sin^3{x}\)
Hint .
Difference of cubes, Pythagorean identity.
Answer .
\((\cos{x} - \sin{x})(1 + \cos{x}\sin{x})\)
Exercise Group 15.1.5 . Advanced Examples.
Simplify each trigonometric expression.
(a) \(\dfrac{\sin^2{x} + \sin{x} - 6}{5\sin{x} + 15}\)
Hint .
Let
\(u=\sin{x}\text{,}\) factor numerator and denominator, cancel.
Answer .
\(\dfrac{\sin{x} - 2}{5}\)
(b) \((\sin{x}+\cos{x})(\sin{x}-\cos{x})+2\cos^2{x}\)
(c) \(\dfrac{\cos^2{x} - 4}{7\cos{x} - 14}\)
Hint .
Difference of squares, factor the denominator, cancel.
Answer .
\(\dfrac{\cos{x} + 2}{7}\)
(d) \(\dfrac{\sin^2{x}\tan{x} - \tan{x}}{\sin{x}\tan{x} + \tan{x}}\)
Hint .
Factor out a common factor, difference of squares, cancel.
Answer .
Subsection 15.1.3 Solving Equations Using Identities
As you learn more fundamental trigonometric identities, you will be able to solve more trigonometric equations algebraically.
Exercise Group 15.1.6 . Solve Each Equation I.
Solve each equation, by finding all solutions in the interval
\([0,2\pi)\text{.}\)
(a) \(2\cos^2{x} - 3 \sin{x} = 0\)
Hint .
Pythagorean identity, leads to
\(2\sin^2{x} + 3\sin{x} - 2 = 0\text{,}\) factor, leads to
\(\sin{x} = \frac{1}{2}\) or
\(\sin{x} = -2\text{.}\)
Answer .
\(x = \frac{\pi}{6}, \frac{5\pi}{6}\text{.}\) General solution:
\(x = \frac{\pi}{6} + 2\pi n, x = \frac{5\pi}{6} + 2\pi n, n \in \mathbb{I}\text{.}\)
(b) \(\cos{x} - 2\sin^2{x} + 1 = 0\)
Hint .
Pythagorean identity, leads to
\(2\cos^2{x} + \cos{x} - 1 = 0\text{,}\) factor, leads to
\(\cos{x} = \frac{1}{2}\) or
\(\cos{x} = -1\text{.}\)
Answer .
\(x = \frac{\pi}{3}, \pi, \frac{5\pi}{3}\text{.}\) General solution:
\(x = \frac{\pi}{3} + 2\pi n, x = \frac{5\pi}{3} + 2\pi n, x = \pi + 2\pi n, n \in \mathbb{I}\text{.}\)
(c) \(\tan^2{x} - 4\sec{x} + 5 = 0\)
Hint .
Pythagorean identity, leads to
\(\sec^2{x} - 4\sec{x} + 4 = 0\text{,}\) factor, leads to
\(\sec{x} = 2\text{.}\)
Answer .
\(x = \frac{\pi}{3}, \frac{5\pi}{3}\text{.}\) General solution:
\(x = \frac{\pi}{3} + 2\pi n, x = \frac{5\pi}{3} + 2\pi n, n \in \mathbb{I}\text{.}\)
(d) \(2\sin^2{x}=1-\cos{x}\)
Hint .
Pythagorean identity, leads to
\(2\cos^2{x}-\cos{x}-1=0\text{,}\) factor.
Answer .
\(x=0,\frac{2\pi}{3},\frac{4\pi}{3}\text{.}\) General solution:
\(x=\frac{2\pi}{3} n, n \in \mathbb{I}\text{.}\)
Exercise Group 15.1.7 . Solve Each Equation II.
Solve each equation, by finding all solutions in the interval
\([0,2\pi)\text{.}\)
(a) \(2\cos^2{x}+\tan^2{x}=2\)
Hint .
Pythagorean identity, leads to
\(2\cos^4{x}-3\cos^2{x}+1=0\text{,}\) factor, leads to
\(\cos^2{x} = 1\) or
\(\cos^2{x}=\frac{1}{2}\text{.}\)
Answer .
\(x=0,\frac{\pi}{4}, \frac{3\pi}{4}, \pi, \frac{5\pi}{4}, \frac{7\pi}{4}\text{.}\)
(b) \(\sin^2{x} + \cot^2{x} - 1 = 0\)
Hint .
Rewrite in terms of sine and cosine, and Pythagorean identity, leads to
\(\sin^4{x} - 2\sin^2{x} + 1 = 0\) which is in quadratic form, factor, leads to
\(\sin{x} = 1\) or
\(\sin{x} = -1\text{.}\)
Answer .
\(x = \frac{\pi}{2}, \frac{3\pi}{2}\text{.}\)
(c) \(\csc^2{x} - \cot{x} - 1 = 0\)
Hint .
Pythagorean identity, leads to
\(\cot^2{x} - \cot{x} = 0\text{,}\) factor, leads to
\(\cot{x} = 0\) or
\(\cot{x} = 1\) (or, convert to sine and cosine, leads to
\(\cos{x}(\sin{x}-\cos{x})=0\) ).
Answer .
\(x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{\pi}{4}, \frac{5\pi}{4}\text{.}\)
(d) \(\dfrac{2\sin^2{x}-1}{\sin{x}}=2\csc^2{x}-1\)
Hint .
Leads to
\(2\sin^3{x}+\sin^2{x}-\sin{x}-2=0\text{,}\) cubic equation, solve by factor by grouping, leads to
\(\sin{x}=1\text{.}\)
Answer .
\(x=\frac{\pi}{2}\text{.}\)
(e) \(\dfrac{1+\sin{x}}{\cos{x}}+\dfrac{\cos{x}}{1+\sin{x}}=4\)
Hint .
Clear denominators by multiplying by the lowest common denominator, Pythagorean identity, factor by grouping, which leads to
\((2\cos{x}-1)(\sin{x}+1)=0\text{.}\)
Answer .
\(x=\frac{\pi}{3},\frac{5\pi}{3}\) (reject
\(x=\frac{3\pi}{2}\) ).
(f) \(\sin^2{(2x)}-\sin{(2x)}-2=0\)
Hint .
Let
\(u=\sin{(2x)}\text{,}\) then
\(u^2-u-2=0\text{.}\)
Answer .
\(x=\frac{3\pi}{4},\frac{7\pi}{4}\text{.}\)
(g) \(6\sin^2{(2x)}-\sin{(2x)}-1=0\)
Hint .
Let
\(u=\sin{(2x)}\text{,}\) then
\(6u^2-u-1=0\text{.}\)
Answer .
\(x=\frac{\pi}{12},\frac{5\pi}{12},1.74,2.97,\frac{13\pi}{12},\frac{17\pi}{12},4.88,6.11\text{.}\)
Subsection 15.1.4 Non-Permissible Values
In algebra, we know that division by zero is undefined. For example,
\(\frac{1}{x}\) is undefined when
\(x = 0\text{.}\) In a similar way, trig expressions can have values where they are undefined.
A
non-permissible value is a value of the variable that makes an expression undefined (sometimes, it is also called a
restriction ). For trigonometric expressions, this happens whenever a denominator equals zero.
Example 15.1.2 . Non-Permissible Values of \(\cot{x}\) .
Find any non-permissible values of
\(\cot{x}\text{.}\)
Using the quotient identity,
\(\cot{x} = \frac{\cos{x}}{\sin{x}}\text{.}\) The denominator is
\(\sin{x}\text{,}\) so we set it equal to zero,
\begin{equation*}
\sin{x} = 0
\end{equation*}
This occurs when
\(x = 0, \pi, 2\pi, \ldots\text{,}\) or in general,
\(x = \pi n\) where
\(n \in \mathbb{I}\text{.}\)
Therefore, the non-permissible values are
\(x \neq \pi n, n \in \mathbb{I}\text{.}\)
To find non-permissible values, basically:
Example 15.1.3 . Non-Permissible Values of \(\dfrac{\sec{x}+\sin{x}}{\tan{x}}\) .
Find any non-permissible values of
\(\dfrac{\sec{x}+\sin{x}}{\tan{x}}\text{.}\)
First, rewrite in terms of sine and cosine:
\begin{equation*}
\frac{\sec{x}+\sin{x}}{\tan{x}} = \frac{\frac{1}{\cos{x}}+\sin{x}}{\frac{\sin{x}}{\cos{x}}}
\end{equation*}
Even though we could try to simplify this, the non-permissible values come from the original expression. The expression is undefined when:
\(\sec{x}\) is undefined, which is when
\(\cos{x} = 0\)
\(\tan{x}\) itself is undefined when
\(\cos{x} = 0\) (again).
\(\tan{x}\) is in a denominator, so we need
\(\tan{x} \neq 0\text{,}\) which means
\(\sin{x} \neq 0\text{.}\)
Setting
\(\cos{x} = 0\) gives
\(x = \frac{\pi}{2} + \pi n\)
Setting
\(\sin{x} = 0\) gives
\(x = \pi n\text{.}\)
Combined, the non-permissible values are
\(x \neq \frac{\pi}{2} n, n \in \mathbb{I}\) (every multiple of
\(\frac{\pi}{2}\) ).
Exercise Group 15.1.8 . Find Non-Permissible Values.
Find any non-permissible values of each expression.
(a) \(\dfrac{\sin{x}}{\tan{x}}\)
Hint .
\(\sin{x} = 0\) or
\(\cos{x} = 0\text{.}\)
Answer .
\(x \neq \frac{\pi}{2} n, n \in \mathbb{I}\text{.}\)
(b) \(\dfrac{\cot{x}}{1-\sin{x}}\)
Hint .
\(\sin{x} = 0\) or
\(\sin{x} = 1\text{.}\)
Answer .
\(x \neq \pi n\) and
\(x \neq \frac{\pi}{2} + 2\pi n, n \in \mathbb{I}\text{.}\)
(c) \(1 - \sec^2{x}\)
Hint . Answer .
\(x \neq \frac{\pi}{2} + \pi n, n \in \mathbb{I}\text{.}\)
(d) \(\dfrac{\tan{x}}{1-\sin{x}}\)
Hint .
\(\cos{x} = 0\) or
\(\sin{x} = 1\text{.}\)
Answer .
\(x \neq \frac{\pi}{2} + \pi n, n \in \mathbb{I}\text{.}\)
(e) \(\dfrac{\csc{x}}{\cos{x}}\)
Hint .
\(\sin{x} = 0\) or
\(\cos{x} = 0\text{.}\)
Answer .
\(x \neq \frac{\pi}{2} n, n \in \mathbb{I}\text{.}\)
(f) \(\sin^2{x} - \cos^2{x}\)
(g) \(\dfrac{\csc{x}}{\cot{x} + \tan{x}}\)
Hint .
\(\sin{x} = 0\) or
\(\cos{x} = 0\text{.}\)
Answer .
\(x \neq \frac{\pi}{2} n, n \in \mathbb{I}\text{.}\)
(h) \(\dfrac{\sec{x}}{\tan{x} + \cot{x}}\)
Hint .
\(\sin{x} = 0\) or
\(\cos{x} = 0\text{.}\)
Answer .
\(x \neq \frac{\pi}{2} n, n \in \mathbb{I}\text{.}\)
(i) \(\dfrac{\tan{x}}{\cos{x} + 1}\)
Hint .
\(\cos{x} = 0\) or
\(\cos{x} = -1\text{.}\)
Answer .
\(x \neq \frac{\pi}{2} + \pi n\) and
\(x \neq \pi + 2\pi n, n \in \mathbb{I}\text{.}\)
(j) \(\tan{x}\)
Hint . Answer .
\(x \neq \frac{\pi}{2} + \pi n, n \in \mathbb{I}\text{.}\)
(k) \(\dfrac{\sec{x}}{\sin{x}}\)
Hint .
\(\cos{x} = 0\) or
\(\sin{x} = 0\text{.}\)
Answer .
\(x \neq \frac{\pi}{2} n, n \in \mathbb{I}\text{.}\)
Subsection 15.1.5 Verifying a Trigonometric Identity by Graphing
Not all equations are identities. Actually, if you write down a random equation with a combination of trig functions, it probably won’t be an identity.
A simple way to verify if an equation is an identity is to graph both sides using a graphing calculator, like
Desmos .
If an equation is an identity, then both sides of the equation will be equal for all values of the variable, and so the graph of each side will coincide (i.e. overlap each other).
If both graphs overlap completely, the equation is an identity.
On the other hand, if the two graphs don’t match at even a single point, then the equation is not an identity.
Example 15.1.5 . Verify by Graphing.
Verify that
\(1 + \cot^2{x} = \csc^2{x}\) using a graphing calculator. Observe how they coincide (overlap):
Desmos graph .
Subsection 15.1.6 “Verifying” Identities
Example 15.1.6 . Verify for a Single Value.
Verify
\(1 - \sin^2{x} = \sin{x} \cos{x} \cot{x}\) for
\(x = \frac{\pi}{4}\text{.}\)
Example 15.1.7 . Is Checking One Value Enough?
Mark claimed that
\(\frac{1}{\cot x}=\tan x\) is an identity. Marcia let
\(x=30°\) and found that both sides of the equation worked out to
\(\frac{1}{\sqrt{3}}\text{.}\) She said that this proves that the equation is an identity. Is Marcia’s reasoning correct? Explain.
Answer: No. Checking only
\(x=30°\) shows the equality holds at one value, but an identity must hold for all
\(x\) in its domain. However, it is true this is an identity, it’s just not proven by checking only one value.
Example 15.1.8 . Verify for \(x=150°\) .
Verify
\(\tan{x}\csc{x}\sec{x}=\sec^2{x}\) for
\(x=150°\text{.}\)
Answer: For
\(x=150°\text{,}\) both sides are
\(\frac{4}{3}\text{.}\)
