Problem 65 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to \(\sin x \cos x\).

Step 1: Rewrite Trigonometric Functions

The first step is to express the reciprocal trigonometric functions (secant and cosecant) in terms of sine and cosine. We have \( \sec x = \frac{1}{\cos x} \) and \( \csc x = \frac{1}{\sin x} \). Substitute these into the left side of the equation: \[ \frac{\sin x + \cos x}{\frac{1}{\cos x} + \frac{1}{\sin x}}. \]

Step 2: Simplify Denominator

To simplify the denominator, combine the fractions by finding a common denominator: \( \frac{1}{\cos x} = \frac{\sin x}{\sin x \cos x} \) and \( \frac{1}{\sin x} = \frac{\cos x}{\sin x \cos x} \). This gives us: \[ \frac{\sin x}{\sin x \cos x} + \frac{\cos x}{\sin x \cos x} = \frac{\sin x + \cos x}{\sin x \cos x}. \]

Step 3: Simplify Expression

Now, substitute the simplified denominator into the original left-hand side: \[ \frac{\sin x + \cos x}{\frac{\sin x + \cos x}{\sin x \cos x}}. \]

Step 4: Cancel Common Terms

Observe that \( \sin x + \cos x \) in the numerator and denominator cancels out, resulting in: \( \sin x \cos x \).

Step 5: Verify the Identity

Upon simplification, both sides of the equation become equal to \( \sin x \cos x \), hence the identity is verified: \( \frac{\sin x + \cos x}{\sec x + \csc x} = \sin x \cos x \).

Key Concepts

Trigonometric Functions

Trigonometric functions are a group of functions that relate the angles of a triangle to the lengths of its sides. They are fundamental in the study of triangles, waves, and oscillations, and have applications in various fields such as physics, engineering, and architecture.

The most basic trigonometric functions include:

• Sine (\( \sin x \)): relates the length of the opposite side to the hypotenuse in a right triangle.
• Cosine (\( \cos x \)): relates the length of the adjacent side to the hypotenuse.
• Tangent (\( \tan x \)): the ratio of sine to cosine, representing the relationship between the opposite and adjacent sides.

These functions are periodic, meaning they repeat their values in regular intervals. Understanding how they interact helps in solving various mathematical problems.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the basic trigonometric functions by taking the reciprocals, hence offering an inverse perspective on the relationships between angles and sides.

The main reciprocal trigonometric functions include:

• Secant (\( \sec x \)): the reciprocal of cosine, defined as \( \sec x = \frac{1}{\cos x} \).
• Cosecant (\( \csc x \)): the reciprocal of sine, defined as \( \csc x = \frac{1}{\sin x} \).
• Cotangent (\( \cot x \)): the reciprocal of tangent, defined as \( \cot x = \frac{1}{\tan x} \).

These functions are especially useful when solving equations and verifying identities. Understanding how to convert these functions into their primary counterparts, sine and cosine, can help streamline complex expressions and verify trigonometric identities efficiently.

Simplifying Expressions

Simplifying expressions in mathematics generally involves reducing complex expressions into simpler and more manageable forms. This process is key in solving equations and verifying identities.

Here’s some advice for simplifying trigonometric expressions:

• Convert all trigonometric terms to sine and cosine where possible, as these are the foundational trigonometric functions.
• Look for common factors and cancel them to simplify the fraction.
• Combine fractions by finding a common denominator, which eases the simplification.
• Use fundamental trigonometric identities, such as \( \sin^2 x + \cos^2 x = 1 \), to reduce expressions.

By following these steps, you can accurately verify whether two expressions are equivalent, as demonstrated in the exercise. Understanding these simplifying techniques allows you to tackle even the most daunting trigonometric equations with confidence.