Problem 67 - Algebra and Trigonometry

Question

Rewrite each expression in terms of the given function or functions. \(\frac{\tan x+\cot x}{\csc x} ; \cos x\)

Step-by-Step Solution

Answer

The expression \(\frac{\tan x+\cot x}{\csc x}\) can be rewritten in terms of \(\cos x\) as \(\cos x(\sin x + \cos x)\).

Step 1: Express all the functions in the origin expression in terms of sine and cosine

We need to replace each function in terms of sine and cosine. The identities are \(\tan x = \frac{\sin x}{\cos x}\), \(\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}\), and \(\csc x = \frac{1}{\sin x}\). Substituting these identities into the given expression, we obtain \(\frac{\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}}{\frac{1}{\sin x}}\).

Step 2: Simplify the expression

Next, we simplify the above expression. This simplification gives: \(\sin x \cdot \cos x + \cos^2 x = \cos x(\sin x + \cos x)\). Thus, the given expression is simplified to \(\cos x(\sin x + \cos x)\).

Step 3: Check the final expression

Finally, we check the derived expression. The expression is in the form of \(\cos x\), as per the requirement, by confirming that the two expressions are equivalent apart from the variable that is requested to be manipulated.

Key Concepts

Sine and Cosine

Understanding the sine and cosine functions is essential for simplifying trigonometric expressions and equations. These functions form the basic building blocks of trigonometry.

• Sine (\(\sin x\)): This function represents the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine (\(\cos x\)): This function signifies the ratio of the adjacent side to the hypotenuse.

Both functions are crucial because they appear in numerous trigonometric identities. For instance, knowing that \(\tan x = \frac{\sin x}{\cos x}\) allows us to express tangent in terms of sine and cosine.

When solving trigonometric equations, expressing complex terms using sine and cosine often simplifies the process. This simplification step is key when trying to rewrite expressions using specific functions, as seen in this exercise.

Tangent and Cotangent

The functions tangent and cotangent are related to sine and cosine but offer different perspectives.

• Tangent (\(\tan x\)): Defined as \(\tan x = \frac{\sin x}{\cos x}\), it represents the ratio of the sine to the cosine of an angle.
• Cotangent (\(\cot x\)): It is the reciprocal of tangent, written as \(\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}\), describing the ratio of the cosine to the sine.

These relationships are vital for transforming trigonometric expressions. By converting tangent and cotangent into sine and cosine, the computations become more manageable. This technique was used in the provided exercise, enabling the simplification of a complex expression.

Cosecant Function

The cosecant function is less common but still plays a significant role in trigonometric transformations.

• Cosecant (\(\csc x\)): This function is the reciprocal of sine, defined as \(\csc x = \frac{1}{\sin x}\)

Cosecant, alongside other reciprocal functions, helps simplify expressions where direct application of sine might not be convenient.

In the original problem, \(\csc x\) was utilized to express the given equation in terms of sine and cosine. Using \(\csc x\) allowed us to rewrite the denominator, leading to further simplifications. Understanding \(\csc x\) accordingly offers greater flexibility in trigonometric equations and identities.