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

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

Simplify cos (𝐴 βˆ’ 𝐡) βˆ’ cos (𝐴 + 𝐡).

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Video Transcript

Simplify cos 𝐴 minus 𝐡 minus cos 𝐴 plus 𝐡.

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

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

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