Video: Simplifying Trigonometric Expression Using Sum and Difference of Angles Identities

Simplify sin(𝐴 + 𝐵) − sin(𝐴 − 𝐵).

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Video Transcript

Simplify sin of 𝐴 plus 𝐵 minus sin of 𝐴 minus 𝐵.

In order to simplify this expression, we firstly recall two of our compound angle identities. Sin of 𝐴 plus 𝐵 is equal to sin 𝐴 cos 𝐵 plus cos 𝐴 sin 𝐵. Sin 𝐴 minus 𝐵, on the other hand, is equal to sin 𝐴 cos 𝐵 minus cos 𝐴 sin 𝐵. Substituting these into our expression gives us sin 𝐴 cos 𝐵 plus cos 𝐴 sin 𝐵 minus sin 𝐴 cos 𝐵 minus cos 𝐴 sin 𝐵.

We can remove the parentheses by multiplying both of the terms inside by negative one. This gives us negative sin 𝐴 cos 𝐵 and positive cos 𝐴 sin 𝐵. We can now group or collect like terms. Sin 𝐴 cos 𝐵 minus sin 𝐴 cos 𝐵 is equal to zero. Cos 𝐴 sin 𝐵 plus cos 𝐴 sin 𝐵 is equal to two cos 𝐴 sin 𝐵. The simplified version of sin 𝐴 plus 𝐵 minus sin 𝐴 minus 𝐵 is two cos 𝐴 sin 𝐵.

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