Question Video: Simplifying Trigonometric Expressions Using the Sum and Difference of Angles Identities Mathematics

Video Transcript

Simplify cos 𝐴 minus 𝐵 minus cos 𝐴 plus 𝐵.

In order to simplify this expression, we firstly need to recall two of our compound-angle identities. Cos 𝐴 minus 𝐵 is equal to cos 𝐴 cos 𝐵 plus sin 𝐴 sin 𝐵. Cos 𝐴 plus 𝐵, on the other hand, is equal to cos 𝐴 cos 𝐵 minus sin 𝐴 sin 𝐵. We can now substitute these identities into our expression. Our expression becomes cos 𝐴 cos 𝐵 plus sin 𝐴 sin 𝐵 minus cos 𝐴 cos 𝐵 minus sin 𝐴 sin 𝐵.

As we’re subtracting the two terms inside the parentheses, this simplifies to become negative cos 𝐴 cos 𝐵 plus sin 𝐴 sin 𝐵. Subtracting a negative term produces a positive term. At this stage, we can then collect or group like terms. Cos 𝐴 cos 𝐵 minus cos 𝐴 cos 𝐵 is equal to zero. Sin 𝐴 sin 𝐵 plus sin 𝐴 sin 𝐵 is equal to two sin 𝐴 sin 𝐵. This is the simplified version of cos 𝐴 minus 𝐵 minus cos 𝐴 plus 𝐵.