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Trigonometric Identities

Trigonometric Equations

Overview Identities Fundamental Sum & Difference Half-Angle Double-Angle Sum & Product

Sum & Product Identities

For any angles or values \(A\) and \(B\), the following relationships are always true.

\[\begin{align*} \sin(A) + \sin(B) &= 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \\[6pt] \sin(A) - \sin(B) &= 2\cos\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right) \end{align*}\]
\[\begin{align*} \cos(A) + \cos(B) &= 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \\[6pt] \cos(A) - \cos(B) &= -2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right) \end{align*}\]

The formulas above are sometimes called the Sum-to-Product Identities because they describe a relationship that let's us convert the sum (or difference) of two sine functions or two cosine functions into a product of sine or cosine functions. Similarly, the formulas below are sometimes called the Product-to-Sum Identities because they describe a relationship that let's us convert the product of two sine or cosine functions into a sum (or difference) of two sine functions or two cosine functions.

\[\begin{align*} \sin(A)\cos(B) &= \frac{1}{2}\left[ \sin(A+B) + \sin(A-B) \right] \\[6pt] \cos(A)\sin(B) &= \frac{1}{2}\left[ \sin(A+B) - \sin(A-B) \right] \end{align*}\]
\[\begin{align*} \sin(A)\sin(B) &= \frac{1}{2}\left[ \cos(A-B) - \cos(A+B) \right] \\[6pt] \cos(A)\cos(B) &= \frac{1}{2}\left[ \cos(A+B) + \cos(A-B) \right] \end{align*}\]

Simplifying Expressions

Let's look at some more examples that use these new identities.

Simplifying Trig Expressions - Sum-Product
[https://youtu.be/Sy9g3_WFx50]