Problem 31 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\cos u \sec u}{\tan u}=\cot u $$

Step-by-Step Solution

Answer

The given identity is verified true.

Step 1: Convert Trigonometric Functions

Rewrite the left side of the equation in terms of sine and cosine. Recall that \( \sec u = \frac{1}{\cos u} \) and \( \tan u = \frac{\sin u}{\cos u} \). Substitute these identities into the expression: \[\frac{\cos u \cdot \frac{1}{\cos u}}{\frac{\sin u}{\cos u}} = \frac{1}{\frac{\sin u}{\cos u}}\]

Step 2: Simplify the Complex Fraction

Simplify the fraction by multiplying the numerator by the reciprocal of the denominator. This gives us: \[\frac{1}{\frac{\sin u}{\cos u}} = \cos u \cdot \frac{1}{\sin u} = \frac{\cos u}{\sin u}\]

Step 3: Recognize the Cotangent

Identify that \( \frac{\cos u}{\sin u} \) is exactly the definition of \( \cot u \). Thus, the expression simplifies to: \[\cot u\]

Step 4: Confirm the Identity

The original equation simplifies to \( \cot u \) on both sides of the equation, confirming that: \[\frac{\cos u \sec u}{\tan u} = \cot u\] This verifies that the identity is true.

Key Concepts

Sine and Cosine

Understanding sine and cosine is fundamental to comprehending trigonometric identities. Sine (\(\sin \)) and cosine (\(\cos\)) relate the angles of a triangle to the lengths of its sides in a right-angled triangle. Here’s a quick rundown of these functions:

• Sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
• Cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.

By using these definitions, we can express many other trigonometric functions. In the given exercise, cosine specifically plays a crucial role as the main term that gets expanded into different identities. By expressing \(\sec u\) as \(\frac{1}{\cos u}\) and \(\tan u\) as \(\frac{\sin u}{\cos u}\), we can simplify more complex trigonometric expressions into basic ones, making it easier to identify the equivalent expression on the simpler side of an equation.

Secant and Cosecant

Secant and cosecant functions are reciprocal identities of cosine and sine, respectively. Understanding them helps in transforming expressions to manipulate equations more easily. Here is what you need to know:

• Secant (\(\sec u\)) is the reciprocal of cosine, meaning \(\sec u = \frac{1}{\cos u}\).
• Cosecant (\(\csc u\)) is the reciprocal of sine, so \(\csc u = \frac{1}{\sin u}\).

Because these functions can turn division into multiplication, they are especially helpful when simplifying the expression of trigonometric identities. For example, using secant allowed the complex fraction in the original exercise to reduce significantly, helping us move from a complicated form to the easier recognition of \(\cot u\). This is crucial for verifying the identity.

Tangent and Cotangent

Tangent and cotangent are closely related trigonometric functions that describe different aspects of a right triangle.

• Tangent (\(\tan u\)) is the ratio of the opposite side to the adjacent side, which can also be expressed as \(\frac{\sin u}{\cos u}\).
• Cotangent (\(\cot u\)) is the reciprocal of tangent, expressed as \(\cot u = \frac{\cos u}{\sin u}\).

In the original exercise, seeing \(\tan u\) transformed and eventually equal to \(\cot u\) confirms our identity. This transformation, utilizing the relationship between tangent and cotangent, simplified the left side of the original equation into a form that exactly matched the expression \(\cot u\) on the right. Understanding these functions and their reciprocals helps in seeing the equivalency in identities and proves the relationships definitively.