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When working with trigonometric identities, one powerful tool is the cosine addition formula. This formula simplifies expressions involving the cosine of the sum or difference of two angles.
\[\cos(a+b) = \cos a \cos b - \sin a \sin b\]
Similarly, for the difference of two angles, the formula is:
\[\cos(a-b) = \cos a \cos b + \sin a \sin b\]
These formulas allow us to express complex trigonometric expressions in terms of basic sine and cosine functions. By doing so, we can more easily identify patterns and simplify expressions, making it easier to solve trigonometric problems. In the original exercise, these formulas helped in expanding \(\cos(a+b)\) and \(\cos(a-b)\) into manageable components.

A trigonometric proof involves showing that two trigonometric expressions are equivalent. The process typically involves using known identities and algebraic manipulations to transform one side of an equation into the other. This kind of proof is a crucial skill in mathematics because it strengthens your understanding of how trigonometric functions relate to each other.
Here’s a simple strategy you might follow for a trigonometric proof:

• Identify known identities that could simplify the expressions.
• Apply these identities to transform one or both sides of the equation.
• Look for opportunities to factor or combine like terms.
• Ensure each step logically follows the previous one, aiming to show equivalency.

In the original identity \( \cos(a+b) + \cos(a-b) = 2\cos a \cos b \), the trigonometric proof relied on expanding each cosine term and then simplifying by combining like terms (canceling out the \(\sin a \sin b\) components), ultimately showing both sides are equivalent.

Angle addition and subtraction are fundamental concepts used in trigonometry to find values of trigonometric functions for non-standard angles.
The angle addition and subtraction formulas are tools that allow you to calculate the trigonometric functions of the sum or difference of two angles. This is crucial because it expands the scope of angles you can deal with using basic trigonometric values.
For instance, using the angle addition formula:
\[\cos(a+b) = \cos a \cos b - \sin a \sin b\]
And the subtraction formula:
\[\cos(a-b) = \cos a \cos b + \sin a \sin b\]
These can be especially helpful when dealing with angles not typically found in basic trigonometric tables, allowing a derivation of values based on known angles.