 Compound Angles: tan(A+B) and tan(A-B)

what you'll learn...

Compound angles of tan & cot

Quickly derive the identities. No need to memorize.

»  $\mathrm{tan}\left(A±B\right)=\frac{\mathrm{sin}\left(A±B\right)}{\mathrm{cos}\left(A±B\right)}$$\tan \left(A \pm B\right) = \frac{\sin \left(A \pm B\right)}{\cos \left(A \pm B\right)}$

»  $\mathrm{cot}\left(A±B\right)=\frac{\mathrm{cos}\left(A±B\right)}{\mathrm{sin}\left(A±B\right)}$$\cot \left(A \pm B\right) = \frac{\cos \left(A \pm B\right)}{\sin \left(A \pm B\right)}$

tan

Proof for tan(A+B) and tan(A-B) using previous results and algebra of trigonometric functions.

$\mathrm{tan}\left(A+B\right)$$\tan \left(A + B\right)$
$\quad \quad = \frac{\sin \left(A + B\right)}{\cos \left(A + B\right)}$
$\quad \quad = \frac{\sin A \cos B + \cos A \sin B}{\cos A \cos B - \sin A \sin B}$
dividing numerator and denominator by $\mathrm{cos}A\mathrm{cos}B$$\cos A \cos B$
$\quad \quad = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

$\mathrm{tan}\left(A-B\right)$$\tan \left(A - B\right)$
$\quad \quad = \tan \left(A + \left(- B\right)\right)$
$\quad \quad = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

summary

$\mathrm{tan}\left(A+B\right)=\frac{\mathrm{tan}A+\mathrm{tan}B}{1-\mathrm{tan}A\mathrm{tan}B}$$\tan \left(A + B\right) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$
$\mathrm{tan}\left(A-B\right)=\frac{\mathrm{tan}A-\mathrm{tan}B}{1+\mathrm{tan}A\mathrm{tan}B}$$\tan \left(A - B\right) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

cot

Proof for cot(A+B) and cot(A-B) using previous results and algebra of trigonometric functions.

$\mathrm{cot}\left(A+B\right)$$\cot \left(A + B\right)$
$\quad \quad = \frac{\cos \left(A + B\right)}{\sin \left(A + B\right)}$
$\quad \quad = \frac{\cos A \cos B - \sin A \sin B}{\sin A \cos B + \cos A \sin B}$
dividing numerator and denominator by $\mathrm{sin}A\mathrm{sin}B$$\sin A \sin B$
$\quad \quad = \frac{\cot A \cot B - 1}{\cot A + \cot B}$

$\mathrm{cot}\left(A-B\right)$$\cot \left(A - B\right)$
$\quad \quad = \cot \left(A + \left(- B\right)\right)$
$\quad \quad = \frac{- \cot A \cot B - 1}{\cot A - \cot B}$

summary

$\mathrm{cot}\left(A+B\right)=\frac{\mathrm{cot}A\mathrm{cot}B-1}{\mathrm{cot}A+\mathrm{cot}B}$$\cot \left(A + B\right) = \frac{\cot A \cot B - 1}{\cot A + \cot B}$
$\mathrm{cot}\left(A-B\right)=\frac{-\mathrm{cot}A\mathrm{cot}B-1}{\mathrm{cot}A-\mathrm{cot}B}$$\cot \left(A - B\right) = \frac{- \cot A \cot B - 1}{\cot A - \cot B}$

summary

Trigonometric Identities for compound Angles

$\mathrm{sin}\left(A+B\right)=\mathrm{sin}A\mathrm{cos}B+\mathrm{cos}A\mathrm{sin}B$$\sin \left(A + B\right) = \sin A \cos B + \cos A \sin B$
$\mathrm{sin}\left(A-B\right)=\mathrm{sin}A\mathrm{cos}B-\mathrm{cos}A\mathrm{sin}B$$\sin \left(A - B\right) = \sin A \cos B - \cos A \sin B$
$\mathrm{cos}\left(A+B\right)=\mathrm{cos}A\mathrm{cos}B-\mathrm{sin}A\mathrm{sin}B$$\cos \left(A + B\right) = \cos A \cos B - \sin A \sin B$
$\mathrm{cos}\left(A-B\right)=\mathrm{cos}A\mathrm{cos}B+\mathrm{sin}A\mathrm{sin}B$$\cos \left(A - B\right) = \cos A \cos B + \sin A \sin B$

$\mathrm{tan}\left(A+B\right)=\frac{\mathrm{tan}A+\mathrm{tan}B}{1-\mathrm{tan}A\mathrm{tan}B}$$\tan \left(A + B\right) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$
$\mathrm{tan}\left(A-B\right)=\frac{\mathrm{tan}A-\mathrm{tan}B}{1+\mathrm{tan}A\mathrm{tan}B}$$\tan \left(A - B\right) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

$\mathrm{cot}\left(A+B\right)=\frac{\mathrm{cot}A\mathrm{cot}B-1}{\mathrm{cot}A+\mathrm{cot}B}$$\cot \left(A + B\right) = \frac{\cot A \cot B - 1}{\cot A + \cot B}$
$\mathrm{cot}\left(A-B\right)=\frac{-\mathrm{cot}A\mathrm{cot}B-1}{\mathrm{cot}A-\mathrm{cot}B}$$\cot \left(A - B\right) = \frac{- \cot A \cot B - 1}{\cot A - \cot B}$

Outline

It is advised to do the firmfunda version of "basics of Trigonometry" course before doing this.

The outline of material to learn "Advanced Trigonometry" is as follows.
Note: go to detailed outline of Advanced Trigonometry

→   Unit Circle form of Trigonmetric Values

→   Trigonometric Values in all Quadrants

→   Trigonometric Values or any Angles : First Principles

→   Understanding Trigonometric Values in First Quadrant

→   Trigonometric Values in First Quadrant

→   Trigonometric Values of Compound Angles: Geometrical Proof

→   Trigonometric Values of Compound Angles: Algebraic Proof

→   Trigonometric Values of Compound Angles: tan cot

→   Trigonometric Values of Compound Angles: more results