Addition and Subtraction Formulas for Tangent and Cotangent
Tangent Addition Formula
On the previous page, we have derived the addition identities for sine and cosine:
\[\sin \left( {\alpha + \beta } \right) = \sin \alpha \cos \beta + \cos \alpha \sin \beta ,\]
\[\cos \left( {\alpha + \beta } \right) = \cos \alpha \cos \beta - \sin \alpha \sin \beta .\]
Suppose now that \(\cos \left( {\alpha + \beta } \right) \ne 0,\) or \(\alpha + \beta \ne \frac{\pi }{2} + \pi n,\) \(n \in \mathbb{Z}.\) Moreover, let also \(\cos \alpha \ne 0\) and \(\cos \beta \ne 0,\) that is, \(\alpha, \beta \ne \frac{\pi }{2} + \pi n,\) \(n \in \mathbb{Z},\) so that we can divide by \(\cos\alpha\cos\beta.\)
Then the tangent addition formula is given by
\[\require{cancel} \tan \left( {\alpha + \beta } \right) = \frac{{\sin \left( {\alpha + \beta } \right)}}{{\cos \left( {\alpha + \beta } \right)}} = \frac{{\sin \alpha \cos \beta + \cos \alpha \sin \beta }}{{\cos \alpha \cos \beta - \sin \alpha \sin \beta }} = \frac{{\frac{{\sin \alpha \cos \beta + \cos \alpha \sin \beta }}{{\cos \alpha \cos \beta }}}}{{\frac{{\cos \alpha \cos \beta - \sin \alpha \sin \beta }}{{\cos \alpha \cos \beta }}}} = \frac{{\frac{{\sin \alpha \cancel{\cos \beta} }}{{\cos \alpha \cancel{\cos \beta} }} + \frac{{\cancel{\cos \alpha} \sin \beta }}{{\cancel{\cos \alpha} \cos \beta }}}}{{\frac{\cancel{\cos \alpha \cos \beta }}{\cancel{\cos \alpha \cos \beta }} - \frac{{\sin \alpha \sin \beta }}{{\cos \alpha \cos \beta }}}} = \frac{{\tan \alpha + \tan \beta }}{{1 - \tan \alpha \tan \beta }}.\]
Hence,
\[\tan\left( {\alpha + \beta} \right) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}\]
Tangent Subtraction Formula
The tangent function is odd:
\[\tan \left( { - \beta } \right) = \frac{{\sin \left( { - \beta } \right)}}{{\cos \left( { - \beta } \right)}} = \frac{{ - \sin \beta }}{{\cos \beta }} = - \tan \beta .\]
Replacing \(\beta \to -\beta\) in the tangent addition formula, we obtain the tangent subtraction formula:
\[\tan \left( {\alpha - \beta } \right) = \frac{{\tan \alpha + \tan \left( { - \beta } \right)}}{{1 - \tan \alpha \tan \left( { - \beta } \right)}} = \frac{{\tan \alpha - \tan \beta }}{{1 + \tan \alpha \tan \beta }}.\]
Thus,
\[\tan\left( {\alpha - \beta} \right) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta}\]
Cotangent Addition Formula
Similarly we can establish the addition identity for cotangent.
Let \(\sin \left( {\alpha + \beta } \right) \ne 0,\) that is, \(\alpha + \beta \ne \pi n,\) \(n \in \mathbb{Z}.\) We also assume that \(\sin\alpha \ne 0\) and \(\sin\beta \ne 0,\) or \(\alpha ,\beta \ne \pi n,\) \(n \in \mathbb{Z},\) so that we can divide by \(\sin\alpha\sin\beta.\)
Then we have
\[\cot \left( {\alpha + \beta } \right) = \frac{{\cos \left( {\alpha + \beta } \right)}}{{\sin \left( {\alpha + \beta } \right)}} = \frac{{\cos \alpha \cos \beta - \sin \alpha \sin \beta }}{{\sin \alpha \cos \beta + \cos \alpha \sin \beta }} = \frac{{\frac{{\cos \alpha \cos \beta - \sin \alpha \sin \beta }}{{\sin \alpha \sin \beta }}}}{{\frac{{\sin \alpha \cos \beta + \cos \alpha \sin \beta }}{{\sin \alpha \sin \beta }}}} = \frac{{\frac{{\cos \alpha \cos \beta }}{{\sin \alpha \sin \beta }} - \frac{\cancel{\sin \alpha \sin \beta }}{\cancel{\sin \alpha \sin \beta }}}}{{\frac{{\cancel{\sin \alpha} \cos \beta }}{{\cancel{\sin \alpha} \sin \beta }} + \frac{{\cos \alpha \cancel{\sin \beta} }}{{\sin \alpha \cancel{\sin \beta} }}}} = \frac{{\cot \alpha \cot \beta - 1}}{{\cot \beta + \cot \alpha }}.\]
We got the following result:
\[\cot\left( {\alpha + \beta} \right) = \frac{\cot\alpha\cot\beta - 1}{\cot\alpha + \cot\beta}\]
The cotangent of the sum of two angles can also be expressed in terms of tangents:
\[\cot\left( {\alpha + \beta} \right) = \frac{1 - \tan\alpha\tan\beta}{\tan\alpha + \tan\beta}\]
Cotangent Subtraction Formula
First we note that the cotangent function is odd:
\[\cot \left( { - \alpha } \right) = \frac{{\cos \left( { - \alpha } \right)}}{{\sin \left( { - \alpha } \right)}} = \frac{{\cos \alpha }}{{ - \sin \alpha }} = - \cot \alpha .\]
Now we can easily derive the cotangent subtraction formula. It is obtained by replacing \(\beta \to -\beta\) in the cotangent addition formula:
\[\cot \left( {\alpha - \beta } \right) = \frac{{\cot \alpha \cot \left( { - \beta } \right) - 1}}{{\cot \alpha + \cot \left( { - \beta } \right)}} = \frac{{ - \cot \alpha \cot \beta - 1}}{{\cot \alpha - \cot \beta }} = \frac{{\cot \alpha \cot \beta + 1}}{{\cot \beta - \cot \alpha }}.\]
So, we have
\[\cot\left( {\alpha - \beta} \right) = \frac{\cot\alpha\cot\beta + 1}{\cot\beta - \cot\alpha}\]
In terms of tangents, the cotangent subtraction formula is given by
\[\cot\left( {\alpha - \beta} \right) = \frac{1 + \tan\alpha\tan\beta}{\tan\alpha - \tan\beta}\]
See solved problems on Page 2.
