91Ó°ÊÓ

The angle addition formulas are essential tools in trigonometry, allowing us to compute the trigonometric functions of angles that are sums or differences of two known angles. The formulas for cosine and sine are expressed as follows:

1. **Cosine Angle Addition Formula**: The formula for adding two angles in cosine is given by \[ \cos (x+y) = \cos x \cos y - \sin x \sin y \] This formula is incredibly useful when breaking down more complex trigonometric expressions into more manageable terms.

2. **Sine Angle Addition Formula**: Similarly, the formula for sine is \[ \sin (x+y) = \sin x \cos y + \cos x \sin y \] By using these formulas, any angle expressed as a sum or difference of other angles can be rewritten, aiding in simplification and calculation.

Cosine is one of the primary trigonometric functions and is denoted by \( \cos \). It represents the x-coordinate of a point on the unit circle corresponding to a given angle. Cosine values are integral in determining angles and distances in various mathematical problems.

Key points about cosine:

• It is an even function, meaning \( \cos(-x) = \cos(x) \).
• Its value ranges between -1 and 1.
• Used in the cosine angle addition formula to simplify expressions.

In the problem above, the angle addition formula for cosine is directly applied to the term \( \cos(B-A) \) transforming the equation into terms more easily matched with the right-hand side of the equation.

The sine function, denoted as \( \sin \), is another fundamental trigonometric function. It measures the y-coordinate of a point on the unit circle corresponding to a given angle. Sine, like cosine, plays an essential role in the analysis of angles and the construction of geometric models.

Key aspects of sine include:

• Sine is an odd function, thus \( \sin(-x) = -\sin(x) \).
• Similar to cosine, its range is from -1 to 1.
• The sine angle addition formula, \( \sin(x+y) \), breaks complex trigonometric problems into simpler components.

Within the solution provided, sine interacts with other terms to further simplify and verify the equation. Notably, \( \sin A \sin(B-A) \) is transformed and manipulated through expansion.

The Pythagorean identity is a cornerstone in trigonometry, deriving from the Pythagorean theorem. This identity states:
\[ \sin^2 x + \cos^2 x = 1 \]
Its reliability arises from the inherent properties of sine and cosine on the unit circle, where any point (cos(x), sin(x)) lies on the circle with radius 1.

Important points about the Pythagorean identity:

• It serves as a check for errors and inconsistencies in calculations.
• Helps to simplify expressions by eliminating variables or consolidating terms.

In the solution to our exercise, the Pythagorean identity simplifies the expression by substituting \( \cos^2A + \sin^2A \) with 1, thereby confirming the equation \( \cos B = \cos B \). This verification step finalizes the solution with clarity and accuracy.