Problem 63 - Algebra and Trigonometry

Question

Verify the identity: $$ \frac{\sec x}{\cot x+\tan x}=\sin x $$

Step-by-Step Solution

Answer

The given identity \( \frac{\sec x}{\cot x + \tan x} = \sin x \) holds true for all x.

Step 1: Rewrite in terms of basic trigonometric functions

Start by rewriting \( \sec x \), \( \cot x \), and \( \tan x \) in terms of basic trigonometric functions. Express secant as \( \frac{1}{\cos x} \), cotangent as \( \frac{\cos x}{\sin x} \), and tangent as \( \frac{\sin x}{\cos x} \). Thus, the left-hand side transforms into \( \frac{1}{\cos x} \times \frac{1}{(\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x})} \).

Step 2: Simplify the expression

Simplify by finding a common denominator for the two fractions in the denominator of the larger fraction. The common denominator is \( \sin x \cos x \), so rewrite the fraction in the denominator as \( \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} \). The left-hand side now becomes \( \frac{1}{\cos x} \times \frac{\sin x \cos x}{\sin^2 x + \cos^2 x} \).

Step 3: Apply the Pythagorean identity

The denominator of the last fraction, \( \sin^2 x + \cos^2 x \), is equal to 1 according to the Pythagorean identity. Therefore, the left-hand side simplifies to \( \frac{1}{\cos x} \times \sin x \cos x \).

Step 4: Simplify even further

In the last fraction, \( \cos x \) appears both in the numerator and the denominator. Therefore, it can be cancelled out, resulting in \( \sin x \), which fulfills the identity.

Key Concepts

Basic Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. In trigonometry, six basic functions play a vital role: sine (\textbf{sin}), cosine (\textbf{cos}), tangent (\textbf{tan}), cosecant (\textbf{csc}), secant (\textbf{sec}), and cotangent (\textbf{cot}).

Each of these functions takes an angle as an input and returns a ratio of two sides of a right triangle. For instance, \( \textbf{sin}(x) \) is the ratio of the length of the side opposite to the angle \( x \) to the length of the hypotenuse. Similarly, \( \textbf{cos}(x) \) is the ratio of the adjacent side to the hypotenuse, and \( \textbf{tan}(x) \) is the ratio of the opposite side to the adjacent side.

Understanding these functions is crucial when dealing with trigonometric identities, as they often require the conversion of one function into another to simplify expressions and verify identities.

Pythagorean Identity

One of the most fundamental ideas in trigonometry is the Pythagorean identity, which states that for any angle \( x \), the square of the sine function plus the square of the cosine function equals one: \[ \textbf{sin}^2(x) + \textbf{cos}^2(x) = 1. \]This equation stems from the Pythagorean theorem, which applies to right-angled triangles.

The Pythagorean identity is incredibly useful when simplifying trigonometric expressions, especially when such expressions involve \( \textbf{sin}^2(x) \) or \( \textbf{cos}^2(x) \). Recognizing that these squares can be swapped with one minus the other square allows for further simplification. This concept is beautifully demonstrated when verifying trigonometric identities.

Trigonometric Simplification

The process of trigonometric simplification often involves a series of steps where complex trigonometric expressions are reduced to simpler or more recognizable forms, typically involving basic trigonometric functions. Simplification can involve factoring, expanding, finding common denominators, and cancelling out terms.

When verifying identities, you might engage in a variety of algebraic maneuvers. These may include combining fractions, multiplying through by conjugates, or strategically using identities like the Pythagorean identity to reduce terms. The ultimate goal is to transform the original complex expression into a simpler or more familiar form that can be easily compared or verified. By breaking down the process into clear, logical steps, we can untangle even the most complicated trigonometric expressions.