91Ó°ÊÓ

The cosine difference identity is a valuable tool in trigonometry that allows us to simplify expressions involving the cosine of two angles. It states that:

• \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

In our exercise, we are given the expression \( \cos 10^{\circ} \cos 80^{\circ} - \sin 10^{\circ} \sin 80^{\circ} \). Notice how it closely resembles the arrangement in the identity.

By recognizing this similarity, we can apply the cosine difference identity by identifying \( A = 80^{\circ} \) and \( B = 10^{\circ} \). This transforms the problem into a much simpler form, allowing us to say \( \cos(80^{\circ} - 10^{\circ}) \), which simplifies to \( \cos 70^{\circ} \).
This step is crucial because it shows how identities can convert complex expressions into more manageable ones, facilitating calculation and understanding.

In trigonometry, finding exact values is essential for accuracy, especially when dealing with standard angles. Standard angles, like 0°, 30°, 45°, 60°, and 90°, have precise trigonometric values. They are usually derived from the unit circle or trigonometric tables.

For our task, once we derived \( \cos 70^{\circ} \) through the cosine difference identity, we needed an exact value. Interestingly, \( \cos 70^{\circ} \) can be rewritten using the co-function identity:

• \( \cos 70^{\circ} = \sin(90^{\circ} - 70^{\circ}) = \sin 20^{\circ} \)

Though \( 20^{\circ} \) is not a standard angle, understanding this transformation helps in better comprehending trigonometric relationships. If needed, approximate calculations or tables can give further assistance in finding specific numeric values.
This method delicately balances exact and approximate calculations as needed.

The addition and subtraction formulas in trigonometry are pivotal tools that streamline the process of evaluating expressions involving trigonometric functions. These formulas help connect angles and their trigonometric functions, making calculations more efficient.

Within these, the subtraction formulas allow us to express a function as the difference of two angles. In our exercise:

• The formula \( \cos(A - B) = \cos A \cos B - \sin A \sin B \) is particularly useful.

By rewriting the original trigonometric expression \( \cos 10^{\circ} \cos 80^{\circ} - \sin 10^{\circ} \sin 80^{\circ} \) into \( \cos(80^{\circ} - 10^{\circ}) \), we witnessed firsthand how subtraction formulas condense the problem, leaving a simple evaluation of \( \cos 70^{\circ} \).
Mastering these formulas is key to easing the complexity of trigonometric problems, paving the way for more straightforward solutions and honing overall mathematical proficiency.