Skip to main content
K12 LibreTexts

3.2.3: Proofs of Trigonometric Identities

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    Convert to sine/cosine, use basic identities, and simplify sides of the equation.

    Verify that \(\dfrac{\sin^2 x}{\tan^2 x}=1−\sin^2 x\).

    Verifying Trigonometric Identities

    Now that you are comfortable simplifying expressions, we will extend the idea to verifying entire identities. Here are a few helpful hints to verify an identity:

    • Change everything into terms of sine and cosine.
    • Use the identities when you can.
    • Start with simplifying the left-hand side of the equation, then, once you get stuck, simplify the right-hand side. As long as the two sides end up with the same final expression, the identity is true.

    Let's verify the following identities.

    1. \(\dfrac{\cot^2 x}{\csc x}=\csc x−\sin x\)

    Rather than have an equal sign between the two sides of the equation, we will draw a vertical line so that it is easier to see what we do to each side of the equation. Start with changing everything into sine and cosine.

    \dfrac{\cot ^{2} x}{\csc x} & \csc x-\sin x \\
    \dfrac{\dfrac{\cos ^{2} x}{\sin ^{2} x}}{\dfrac{1}{\sin x}} & \dfrac{1}{\sin x} \\
    \dfrac{\cos ^{2} x}{\sin x} &

    Now, it looks like we are at an impasse with the left-hand side. Let’s combine the right-hand side by giving them same denominator.

    \dfrac{1}{\sin x}-\dfrac{\sin ^{2} x}{\sin x} \\
    \dfrac{1-\sin ^{2} x}{\sin x} \\
    \dfrac{\cos ^{2} x}{\sin x}

    The two sides reduce to the same expression, so we can conclude this is a valid identity. In the last step, we used the Pythagorean Identity, \(\sin^2 \theta +\cos^2 \theta =1\), and isolated the \(\cos^2 x=1−\sin^2x\).

    There are usually more than one way to verify a trig identity. When proving this identity in the first step, rather than changing the cotangent to\(\dfrac{\cos^2 x}{\sin^2 x}\), we could have also substituted the identity \(\cot^2 x=\csc^2 x−1\).

    1. \(\dfrac{\sin x}{1−\cos x}=\dfrac{1+\cos x}{\sin x}\)

    Multiply the left-hand side of the equation by \(\dfrac{1+\cos x}{1+\cos x}\).

    \dfrac{\sin x}{1-\cos x} &=\dfrac{1+\cos x}{\sin x} \\
    \dfrac{1+\cos x}{1+\cos x} \cdot \dfrac{\sin x}{1-\cos x} &=\\
    \dfrac{\sin (1+\cos x)}{1-\cos ^{2} x} &=\\
    \dfrac{\sin (1+\cos x)}{\sin ^{2} x} &=\\
    \dfrac{1+\cos x}{\sin x} &=

    The two sides are the same, so we are done.

    1. \(\sec(−x)=\sec x\)

    Change secant to cosine.


    From the Negative Angle Identities, we know that \(\cos(−x)=\cos x\).

    \(\begin{aligned} &=\dfrac{1}{\cos x} \\&=\sec x\end{aligned}\)

    Example \(\PageIndex{1}\)

    Earlier, you were asked to verify that \(\sin^2 x \tan^2 x=1−\sin^2 x\).


    Start by simplifying the left-hand side of the equation.

    \(\sin ^2 x \tan ^2 x=\dfrac{\sin ^2 x}{\dfrac{\sin ^2 x}{\cos ^2 x}}=\cos ^2 x\)

    Now simplify the right-hand side of the equation. By manipulating the Trigonometric Identity,

    \(\sin ^2 x+\cos ^2 x=1\), we get \(\cos ^2 x=1−\sin ^2 x\).

    \(\cos ^2 x=\cos ^2 x\) and the equation is verified.

    Verify the following identities.

    Example \(\PageIndex{2}\)

    \(\cos x \sec x=1\)


    Change secant to cosine.

    \(\begin{aligned} \cos x \sec x&=\cos x \cdot \dfrac{1}{\cos x} \\&=1\end{aligned}\)

    Example \(\PageIndex{3}\)

    \(2−sec^2 x=1−\tan ^2 x\)


    Use the identity \(1+\tan^2 \theta =\sec^2 \theta\).

    \(\begin{aligned} 2−\sec^2 x &=2−(1+\tan ^2 x) \\&=2−1−\tan ^2 x \\&=1−\tan ^2 x\end{aligned}\)

    Example \(\PageIndex{4}\)

    \(\dfrac{\cos(−x)}{1+\sin(−x)}=\sec x+\tan x\)


    Here, start with the Negative Angle Identities and multiply the top and bottom by \(\dfrac{1+\sin x}{1+\sin x}\) to make the denominator a monomial.

    \dfrac{\cos (-x)}{1+\sin (-x)} &=\dfrac{\cos x}{1-\sin x} \cdot \dfrac{1+\sin x}{1+\sin x} \\
    &=\dfrac{\cos x(1+\sin x)}{1-\sin ^{2} x} \\
    &=\dfrac{\cos x(1+\sin x)}{\cos ^{2} x} \\
    &=\dfrac{1+\sin x}{\cos x} \\
    &=\dfrac{1}{\cos x}+\dfrac{\sin x}{\cos x} \\
    &=\sec x+\tan x


    Verify the following identities.

    1. \(\cot(−x)=−\cot x\)
    2. \(\csc(−x)=−\csc x\)
    3. \(\tan x \csc x \cos x=1\)
    4. \(\sin x+\cos x \cot x=\csc x\)
    5. \(\csc\left(\dfrac{\pi}{2}−x\right)=\sec x\)
    6. \(\tan\left(\dfrac{\pi}{2}−x\right)=\tan x\)
    7. \(\dfrac{\csc x}{\sin x}−\dfrac{\cot x}{\tan x}=1\)
    8. \(\dfrac{\tan ^2 x}{\tan ^2 x+1}=\sin ^2 x\)
    9. \((\sin x−\cos x)^2+(\sin x+\cos x)^2=2\)
    10. \(\sin x−\sin x \cos ^2 x=\sin ^3 x\)
    11. \(\tan ^2 x+1+\tan x \sec x=\dfrac{1+\sin x}{\cos ^2 x}\)
    12. \(\cos ^2 x=\csc x \cos x \tan x+\cot x\)
    13. \(\dfrac{1}{1−\sin x}−\dfrac{1}{1+\sin x}=2\tan x \sec x\)
    14. \(\csc^4 x−\cot^4 x=\csc ^2 x+\cot^2x\)
    15. \((\sin x−\tan x)(\cos x−\cot x)=(\sin x−1)(\cos x−1)\)

    Answers for Review Problems

    To see the Review answers, open this PDF file and look for section 14.9.

    This page titled 3.2.3: Proofs of Trigonometric Identities is shared under a CK-12 license and was authored, remixed, and/or curated by CK-12 Foundation via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

    CK-12 Foundation
    CK-12 Foundation is licensed under CK-12 Curriculum Materials License