Prerequisites

Angle addition for cosine | \(\cos(\theta + \phi) = \cos(\theta) \cdot \cos(\phi) - \sin(\theta) \cdot \sin(\phi)\)

Angle subtraction for cosine | \(\cos(\theta - \phi) = \cos(\theta) \cdot \cos(\phi) + \sin(\theta) \cdot \sin(\phi)\)

Description

Description coming soon...

Equation

\[\htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta}) \cdot \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-2angle}{\phi}) = \frac{1}{2}(\htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta} + \htmlId{tooltip-2angle}{\phi}) + \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta} - \htmlId{tooltip-2angle}{\phi}))\]

Symbols Used

\(\theta\)	This is a commonly used symbol to represent an angle in mathematics and physics.
\(\phi\)	This symbol means the same as Angle, which uses \(\htmlId{tooltip-angle}{\theta}\). It is a secondary symbol to use that represents an angle, when a different angle is already using \(\htmlId{tooltip-angle}{\theta}\).
\(\cos\)	This is the symbol for cosine, a trigonometric function that calculates the ratio of the adjacent side to the hypotenuse of a right-angled triangle.

Derivation

Consider the angle addition for cosine identity:
\[\htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta} + \htmlId{tooltip-2angle}{\phi}) = \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta}) \cdot \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-2angle}{\phi}) - \htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-angle}{\theta}) \cdot \htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-2angle}{\phi})\]
and the angle subtraction for cosine identity:
\[\htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta} - \htmlId{tooltip-2angle}{\phi}) = \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta}) \cdot \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-2angle}{\phi}) + \htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-angle}{\theta}) \cdot \htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-2angle}{\phi})\]
We can now add these equations together to get...
\[\htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta} + \htmlId{tooltip-2angle}{\phi}) + \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta} - \htmlId{tooltip-2angle}{\phi}) = \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta}) \cdot \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-2angle}{\phi}) + \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta}) \cdot \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-2angle}{\phi}) - \htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-angle}{\theta}) \cdot \htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-2angle}{\phi}) + \htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-angle}{\theta}) \cdot \htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-2angle}{\phi})\]
From here, the cosine terms on the right hand side are identical so can be combined, and the sine terms are identical but of opposite sign so can be cancelled out. We therefore simplify to get...
\[\htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta} + \htmlId{tooltip-2angle}{\phi}) + \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta} - \htmlId{tooltip-2angle}{\phi}) = 2 \cdot \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta}) \cdot \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-2angle}{\phi})\]
Finally, we can divide both sides by \(2\) and swap around the order of the left hand side and right hand side to get...
\[\htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta}) \cdot \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-2angle}{\phi}) = \frac{1}{2}(\htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta} + \htmlId{tooltip-2angle}{\phi}) + \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta} - \htmlId{tooltip-2angle}{\phi}))\]
as required.

Example