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In trigonometry, angle addition formulas are crucial tools that help us work efficiently with expressions involving sums or differences of angles. The most frequently used angle addition formula for cosine is

• \(\cos(a + b) = \cos a \cos b - \sin a \sin b\)

Similarly, for the difference of angles, the formula is

• \(\cos(a - b) = \cos a \cos b + \sin a \sin b\)

These identities allow us to break down complex algebraic expressions into simpler ones, enabling easy manipulation and simplification of the terms.
By applying these formulas, we transform the given expression in the exercise from a cumbersome form to a workable format that can be easily simplified.
Such techniques are fundamental not only in solving textbook exercises but also in understanding wave functions, oscillations, and many physics-related computations.

Simplifying expressions in trigonometry involves reducing complex expressions to their simplest form. It often requires combining like terms, using identities, and executing arithmetic operations carefully.
In the problem at hand, after applying the angle addition formulas for cosine, the next step is to combine and simplify the expression:

• \(\cos a \cos b - \sin a \sin b - (\cos a \cos b + \sin a \sin b)\)

By distributing the negative sign and combining like terms, we obtain:

• \(-\cos a \cos b + \sin a \sin b - \cos a \cos b - \sin a \sin b\)

This simplifies further to:

• \(-2 \cos a \cos b + 2 \sin a \sin b\)

Simplifying expressions is critical for clearer insights and for easing further calculations, especially in trigonometric proofs and problem-solving.

Factorization in trigonometry refers to rewriting expressions as products of simpler expressions or terms. This is very useful in solving equations and simplifying solutions further.
In our example, after simplification, the expression was reduced to

• \(-2 \cos a \cos b + 2 \sin a \sin b\)

To factor this expression, we notice the common factor '2' from both terms. Extracting 2 from the expression, we arrive at:

• \(2 (\sin a \sin b - \cos a \cos b)\)

Factoring not only reduces the complexity of expressions but also nicely reveals any inherent properties or symmetries in the trigonometric functions. It is a necessary step to finalize solutions and is particularly valuable when solving trigonometric equations or optimizing expressions for integration in calculus.