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.