Proof of Compound Angle Formula cos(A-B)
The compound angle formula-html-341f8a26.gif){kind=small}
can be proved by considering the sector of the unit circle between angles from the x – axis between A and B.
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The coordinates of-html-33a2c479.gif){kind=small}
are-html-m3f1709d2.gif){kind=small}
and the coordinates of L are-html-m418b577a.gif){kind=small}
The distance,-html-m1c4b065d.gif){kind=small}
betweenand-html-m20d25559.gif){kind=small}
satisfies-html-324f264f.gif){kind=small}
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From the Cosine Rule
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Since the circle is a unit circle,-html-m2b819925.gif){kind=small}
and-html-6720b0b2.gif){kind=small}
so
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Equating these two expressions gives
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which simplifies to-html-m623f1145.gif){kind=small}