Problem 31 - Contemporary Precalculus

Question

In Exercises \(30-36,\) perform the indicated operations, then simplify your answers by using appropriate definitions and identities. $$\cos t \sin t(\csc t+\sec t)$$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\cos t \sin t(\csc t+\sec t)\). Answer: The simplified expression is \(\cos t + \sin t\).

Step 1: Write the reciprocal identities for \(\csc t\) and \(\sec t\)

The reciprocal identities for \(\csc t\) and \(\sec t\) are: $$\csc t = \frac{1}{\sin t}$$ $$\sec t = \frac{1}{\cos t}$$ These identities will help us simplify the given expression.

Step 2: Substitute the reciprocal identities into the given expression

Now we substitute the reciprocal identities from step 1 into the given expression: $$\cos t \sin t(\csc t+\sec t) = \cos t \sin t\left(\frac{1}{\sin t}+\frac{1}{\cos t}\right)$$

Step 3: Distribute \(\cos t \sin t\) to the terms inside the parenthesis

Next, distribute \(\cos t \sin t\) to each term inside the parenthesis: $$\cos t \sin t\left(\frac{1}{\sin t}+\frac{1}{\cos t}\right) = \cos t \sin t \cdot \frac{1}{\sin t} + \cos t \sin t \cdot \frac{1}{\cos t}$$

Step 4: Simplify the expression

Now, simplify the expression by canceling out the common factors: $$\cos t \sin t \cdot \frac{1}{\sin t} + \cos t \sin t \cdot \frac{1}{\cos t} = \frac{\cos t \sin t}{\sin t} + \frac{\cos t \sin t}{\cos t}$$ $$= \cos t + \sin t$$ So, the simplified expression is: $$\cos t \sin t(\csc t+\sec t) = \cos t + \sin t$$.

Key Concepts

Reciprocal Identities

Understanding reciprocal identities is one of the foundational blocks in simplifying trigonometric expressions. Trigonometric functions have pairs that are 'reciprocal' to one another. This means one function is the inverse of another function. Specifically, in the context of our exercise involving \(\cos t \sin t(\csc t+\sec t)\), we use the identities:
\[\csc t = \frac{1}{\sin t}\] and
\[\sec t = \frac{1}{\cos t}\].
When we encounter a complex expression like the one in our example, we can replace \(\csc t\) and \(\sec t\) with their corresponding reciprocal identities. This allows us to simplify the expression to a form that is much easier to work with. It's like having the key to unlock a tangled puzzle. In algebra, this is akin to recognizing and substituting reciprocal numbers, like replacing \(\frac{1}{5}\) with 5.

Trigonometric Identities

Trigonometric identities are the equations that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equation are defined. These identities are crucial in simplifying trigonometric expressions and solving trigonometric equations. In our problem, using the reciprocal identities is part of this larger set of tools. Simplifying \(\cos t \sin t(\csc t+\sec t)\) to \(\cos t + \sin t\) not only requires the knowledge of reciprocal identities but also requires distributive properties—a concept we meet in algebra. By understanding how these identities work, one can break down complex trigonometric problems into simpler parts, which can be manipulated and then solved or simplified.

Precalculus Algebra

In precalculus algebra, we delve into more complex operations than in basic algebra, including the manipulation of trigonometric expressions. It lays the groundwork for calculus, as we deal with functions, equations, and identities. In the context of our example, after substituting the reciprocal identities, we applied the distributive property:
\[\cos t \sin t \left(\frac{1}{\sin t}+\frac{1}{\cos t}\right) = \cos t \sin t \cdot \frac{1}{\sin t} + \cos t \sin t \cdot \frac{1}{\cos t}\].
Later, we simplified the expression by canceling out common factors, this step further emphasizes the role of algebra in trigonometry. These algebraic concepts are not only essential for simplifying expressions but also for solving equations, modeling situations, and in the analysis of functions as we advance towards calculus.