Problem 64 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\sec x+\csc x}{\tan x+\cot x}=\sin x+\cos x $$

Step-by-Step Solution

Answer

The identity is verified: \( \frac{\sec x + \csc x}{\tan x + \cot x} = \sin x + \cos x \).

Step 1: Rewrite Trigonometric Functions

We start by rewriting the functions in terms of sine and cosine. The secant and cosecant functions can be rewritten as:\[\sec x = \frac{1}{\cos x}, \quad \csc x = \frac{1}{\sin x}\]The tangent and cotangent functions can be rewritten as:\[\tan x = \frac{\sin x}{\cos x}, \quad \cot x = \frac{\cos x}{\sin x}\]

Step 2: Replace Functions in the Expression

Substitute the rewritten trigonometric functions into the left side of the identity:\[\frac{\sec x + \csc x}{\tan x + \cot x} = \frac{\frac{1}{\cos x} + \frac{1}{\sin x}}{\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}}\]

Step 3: Simplify the Numerator

Combine the terms in the numerator by finding a common denominator:\[\frac{1}{\cos x} + \frac{1}{\sin x} = \frac{\sin x + \cos x}{\sin x \cos x}\]

Step 4: Simplify the Denominator

Combine the terms in the denominator by finding a common denominator:\[\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x}\]Recall the trigonometric identity \(\sin^2 x + \cos^2 x = 1\), so the denominator simplifies to:\[\frac{1}{\sin x \cos x}\]

Step 5: Simplify Entire Fraction

Now substitute the simplified numerator and denominator back into the expression:\[\frac{\frac{\sin x + \cos x}{\sin x \cos x}}{\frac{1}{\sin x \cos x}}\]This simplifies to:\[\sin x + \cos x\]

Step 6: Verify and Conclude

We have simplified the left side of the identity to \(\sin x + \cos x\), which matches the right side of the original identity. Thus, the identity is verified.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially for understanding patterns and relationships within geometry. These functions help relate the angles and sides of triangles, and extend to a circle's properties. In this exercise, we work with four crucial trigonometric functions: secant (\(\sec x\)), cosecant (\(\csc x\)), tangent (\(\tan x\)), and cotangent (\(\cot x\)). Each of these functions can be expressed using the more familiar sine (\(\sin\)) and cosine (\(\cos\)) functions:

• Secant is the reciprocal of cosine: \(\sec x = \frac{1}{\cos x}\)
• Cosecant is the reciprocal of sine: \(\csc x = \frac{1}{\sin x}\)
• Tangent is the ratio of sine to cosine: \(\tan x = \frac{\sin x}{\cos x}\)
• Cotangent is the ratio of cosine to sine: \(\cot x = \frac{\cos x}{\sin x}\)

Knowing these relationships allows us to simplify complex expressions and verify trigonometric identities.

Sine and Cosine

Sine and cosine are the core components of trigonometric functions. They describe ratios related to the angles in right-angled triangles. In this exercise, rewriting each function in terms of sine and cosine is essential for simplification. Sine represents the y-coordinate or vertical component, while cosine represents the x-coordinate or horizontal component on the unit circle.

• The foundational identity, \(\sin^2 x + \cos^2 x = 1\), plays a pivotal role in most trigonometric verifications.
• Understanding the geometric meaning of sine and cosine can help visualize how these functions operate in various contexts, including oscillations and wave forms.

To simplify expressions or verify identities, converting other trigonometric functions into sine and cosine often reveals the simplest form.

Identity Verification

Verifying trigonometric identities involves demonstrating that one side of an equation is equivalent to the other by using known identities and simplifications. This exercise focuses on the identity: \(\frac{\sec x + \csc x}{\tan x + \cot x} = \sin x + \cos x\). We start by expressing all terms in terms of sine and cosine for consistency.

• The numerator becomes \(\frac{\sin x + \cos x}{\sin x \cos x}\) after finding a common denominator.
• The denominator simplifies to \(\frac{1}{\sin x \cos x}\) using the identity \(\sin^2 x + \cos^2 x = 1\).

Simplifying the overall fraction shows that the left-hand side equals the right-hand side, confirming the identity. Such exercises are crucial for mastering the manipulation and simplification skills needed in more advanced trigonometry.