Question Video: Proving the Addition Formula for Sine from its Subtraction Formula | Nagwa Question Video: Proving the Addition Formula for Sine from its Subtraction Formula | Nagwa

Question Video: Proving the Addition Formula for Sine from its Subtraction Formula Mathematics • Second Year of Secondary School

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Using the relation sin (𝛼 − 𝛽) = sin 𝛼 cos 𝛽 − cos 𝛼 sin 𝛽, find an expression for sin (𝛼 + 𝛽).

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Video Transcript

Using the relation sin of 𝛼 minus 𝛽 is equal to sin 𝛼 cos 𝛽 minus cos 𝛼 sin 𝛽, find an expression for sin of 𝛼 plus 𝛽.

In this question, we are asked to use one of the trigonometric difference identities to prove one of the angle sum identities. We are told that sin of 𝛼 minus 𝛽 is equal to sin 𝛼 cos 𝛽 minus cos 𝛼 sin 𝛽. If we replace 𝛽 with negative 𝛽 on the left-hand side, we have sin of 𝛼 minus negative 𝛽. The right-hand side becomes sin 𝛼 multiplied by cos negative 𝛽 minus cos 𝛼 multiplied by sin of negative 𝛽. The expression on the left-hand side can be simplified to sin of 𝛼 plus 𝛽, and we can use our knowledge of odd and even functions to simplify the right-hand side.

Since the cosine function is even, the cos of negative 𝜃 is equal to the cos of 𝜃. This means that cos of negative 𝛽 is equal to cos 𝛽. The sine function is odd; therefore, the sin of negative 𝜃 is equal to negative sin 𝜃. This means that the right-hand side of our equation simplifies to sin 𝛼 multiplied by cos 𝛽 minus cos 𝛼 multiplied by negative sin 𝛽. This, in turn, is equal to sin 𝛼 cos 𝛽 plus cos 𝛼 sin 𝛽, which is an expression for sin of 𝛼 plus 𝛽, which is one of the angle sum identities.

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