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Sum-to-Product Identities

Sometimes we may need to simplify a trigonometric expression like \(\sin \alpha \pm \sin \beta \) or \(\cos \alpha \pm \cos \beta \) by converting the sum or difference of trigonometric functions into a product. These formulas are known as sum-to-product identities.

Sum and Difference of Sines

To derive the sum-to-product identities for sine we use the sine addition and subtraction formulas:

\[\sin \left( {x + y} \right) = \sin x \cos y + \cos x \sin y ,\]
\[\sin \left( {x - y} \right) = \sin x \cos y - \cos x \sin y .\]

If we add and subtract the two equations, we get

\[\sin \left( {x + y} \right) + \sin \left( {x - y} \right) = 2\sin x \cos y,\]
\[\sin \left( {x + y} \right) - \sin \left( {x - y} \right) = 2\cos x \sin y.\]

Let now \(x = y = \alpha\) and \(x - y = \beta.\) Solving this system for \(x\) and \(y,\) we obtain

\[x = \frac{{\alpha + \beta }}{2},\;\;y = \frac{{\alpha - \beta }}{2}.\]

Plugging this into the equations above yields:

Sum of sines
Difference of sines

Sum and Difference of Cosines

Similarly, using the cosine addition and subtraction formulas

\[\cos \left( {x + y} \right) = \cos x\cos y - \sin x\sin y,\]
\[\cos \left( {x - y} \right) = \cos x\cos y + \sin x\sin y,\]

we can derive the sum-to-product identities for cosine:

Sum of cosines
Difference of cosines

Sum and Difference of Tangents

We have

\[\tan \alpha + \tan \beta = \frac{{\sin \alpha }}{{\cos \alpha }} + \frac{{\sin \beta }}{{\cos \beta }} = \frac{{\sin \alpha \cos \beta + \cos \alpha \sin \beta }}{{\cos \alpha \cos \beta}}.\]

The numerator in the last expression is equal to \(\sin \left( {\alpha + \beta } \right).\) Hence,

Sum of tangents

This formula is valid if \(\cos\alpha \ne 0\) and \(\cos\beta \ne 0,\) or \(\alpha ,\beta \ne \frac{\pi }{2} + \pi n,\) \(n \in \mathbb{Z}.\)

Consider the difference of tangents:

\[\tan \alpha - \tan \beta = \frac{{\sin \alpha }}{{\cos \alpha }} - \frac{{\sin \beta }}{{\cos \beta }} = \frac{{\sin \alpha \cos \beta - \cos \alpha \sin \beta }}{{\cos \alpha \cos \beta }} = \frac{{\sin \left( {\alpha - \beta } \right)}}{{\cos \alpha \cos \beta }},\]

that is,

Difference of tangents

Sum and Difference of Cotangents

The sum-to-product formulas for cotangent can de derived in the same way. They are given by

Sum of cotangents
Difference of cotangents

where \(\sin\alpha \ne 0\) and \(\sin\beta \ne 0,\) or \(\alpha ,\beta \ne \pi n,\) \(n \in \mathbb{Z}.\)

See solved problems on Page 2.

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