## Sum-Difference Rule

### The sum-difference rule is a mathematical rule or formula that allows us to simplify expressions involving the sum or difference of two terms or functions

The sum-difference rule is a mathematical rule or formula that allows us to simplify expressions involving the sum or difference of two terms or functions. It applies to both algebraic expressions and trigonometric functions.

For algebraic expressions, the sum-difference rule states:

(a + b)(c + d) = ac + ad + bc + bd

This rule expands the product of two binomial expressions (a + b) and (c + d) by distributing and combining like terms.

For example, if we have the expression (2x + 3)(4x – 5), we can use the sum-difference rule to find its expanded form:

(2x + 3)(4x – 5) = 2x * 4x + 2x * (-5) + 3 * 4x + 3 * (-5)

= 8x^2 – 10x + 12x – 15

= 8x^2 + 2x – 15

In trigonometry, the sum-difference rule applies to trigonometric functions. It allows us to simplify expressions involving the sum or difference of two trigonometric functions.

For example, if we have the expression sin(a + b), we can use the sum-difference rule to simplify it:

sin(a + b) = sin(a) * cos(b) + cos(a) * sin(b)

This rule allows us to express the sine of a sum of angles in terms of the sines and cosines of the individual angles.

Similarly, for the difference of angles, we have the formula:

sin(a – b) = sin(a) * cos(b) – cos(a) * sin(b)

These trigonometric sum-difference formulas are useful in simplifying and solving trigonometric equations or expressions involving multiple angles.

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