91Ó°ÊÓ

Trigonometric identities are fundamental tools in simplifying trigonometric expressions. They establish relationships between different trigonometric functions. These identities include well-known formulas that help us manipulate and transform expressions into more workable forms. In our given exercise, the identity used is \[\tan \theta = \frac{\sin \theta}{\cos \theta}\], which relates tangent to sine and cosine.
When working with trigonometric identities, it’s crucial to recognize which identities apply to the given problem. Identifying the correct identity allows you to replace terms in expressions with their equivalent forms. This can often simplify complex trigonometric expressions into something more manageable. Knowing a variety of identities not only eases calculations but also enhances understanding of trigonometric relationships.

The substitution method involves replacing one part of an expression with an equivalent value, often to simplify or solve the expression. In trigonometry, substitution is frequently used in tandem with trigonometric identities to make expressions simpler to understand and solve.
For the exercise we're considering, the substitution method involves taking our identity, \[\tan \theta = \frac{\sin \theta}{\cos \theta}\], and substituting \( \tan \theta \) with \( \frac{\sin \theta}{\cos \theta} \) in the original expression \( \cos \theta + \sin \theta \tan \theta \).
This approach offers a clearer path to breaking down complex expressions, as it allows us to work with simpler terms. Substitution is particularly useful for revealing hidden simplifications that are not immediately obvious.

Simplification of expressions is the process of breaking down complex expressions into simplest forms while maintaining their value. This is achieved through strategic operations and manipulations, such as using identities or performing arithmetic operations, like addition, subtraction, or factoring.
In our case, after applying the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we substitute it into the expression to obtain \( \cos \theta + \sin \theta \cdot \frac{\sin \theta}{\cos \theta} \).
The next step in simplification involves performing the multiplication, leading us to \( \cos \theta + \sin^2 \theta \).
The act of simplification not only makes the expression easier to work with but also helps in identifying further operations or transformations, if necessary. This process is crucial for solving equations, proving identities, and making calculations manageable.