Problem 67 - 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 as both sides simplify to \( \sin x \cos x \).

Step 1: Rewrite the Secant and Cosecant Terms

Recall that secant and cosecant can be rewritten in terms of sine and cosine:\[\sec x = \frac{1}{\cos x}, \quad \csc x = \frac{1}{\sin x}\]Thus, the denominator \( \sec x + \csc x \) can be rewritten as:\[\frac{1}{\cos x} + \frac{1}{\sin x} = \frac{\sin x + \cos x}{\sin x \cos x}\]

Step 2: Simplify the Fraction

The original expression is:\[\frac{\sin x + \cos x}{\sec x + \csc x}\]Substitute the expression for \( \sec x + \csc x \) from Step 1:\[\frac{\sin x + \cos x}{\frac{\sin x + \cos x}{\sin x \cos x}}\]

Step 3: Simplify Using Division Properties

Simplify the expression by multiplying by the reciprocal of the denominator:\[(\sin x + \cos x) \div \left( \frac{\sin x + \cos x}{\sin x \cos x} \right) = (\sin x + \cos x) \times \left( \frac{\sin x \cos x}{\sin x + \cos x} \right)\]Canceling \( \sin x + \cos x \) from the numerator and denominator yields:\[\sin x \cos x\]

Step 4: Verify the Final Expression

After canceling, we are left with \( \sin x \cos x \) on the right side, which matches the expression the identity claims:\[\frac{\sin x+\cos x}{\sec x+\csc x} = \sin x \cos x\]Thus, the identity is verified.

Key Concepts

Sine and Cosine Functions

Sine and cosine are foundational trigonometric functions. They relate to the coordinates on the unit circle.
Sine (\( ext{sin}\)) measures the vertical distance from the x-axis to the circle's edge, while cosine (\( ext{cos}\)) measures the horizontal distance.
Their values oscillate between -1 and 1 as an angle rotates around the circle.

• Properties: They are periodic functions, repeating every \(2\pi\).
  The basic identities include \( ext{sin}^2 x + ext{cos}^2 x = 1\).
• Graphical Representation: Sine and cosine functions create smooth, wave-like graphs, known as sinusoids.
  This visualization helps in understanding their behavior over different cycles.

Understanding these functions is crucial for solving trigonometric identities, such as verifying the \(\frac{\sin x + \cos x}{\sec x + \csc x} = \sin x \cos x\).
By rewriting terms to involve sine and cosine, many complex expressions become more manageable.

Secant and Cosecant

Secant (\( ext{sec}\)) and cosecant (\( ext{csc}\)) are reciprocals of cosine and sine, respectively.
They are less commonly used but are essential for solving certain problems, including the exercise here.

• Definitions: \( ext{sec} x = \frac{1}{\cos x}\) and \( ext{csc} x = \frac{1}{\sin x}\).
• Properties: Unlike sine and cosine, the ranges of secant and cosecant do not exist between -1 and 1.
  Instead, their values are outside this range, either less than -1 or greater than 1.

For the given trigonometric identity, rewriting secant and cosecant in terms of sine and cosine simplifies the problem.
It transforms the denominator from \(\sec x + \csc x\) into a fraction that can be easily approached using algebraic manipulation.
This step is vital in simplifying expressions in trigonometry.

Algebraic Manipulation in Trigonometry

Algebraic manipulation involves using algebraic operations to simplify or transform expressions.
It is a crucial skill in trigonometry, allowing us to verify identities like the one in the exercise.

• Simplification Process: Begin by substituting equivalent expressions, such as using \(\text{sec} x = \frac{1}{\cos x}\) and \(\text{csc} x = \frac{1}{\sin x}\).
• Cancellation: After rewriting, check for common terms in the numerator and denominator, then simplify further by canceling those terms.

In our exercise, rewrite the denominator to involve sine and cosine.
Then, multiply by the reciprocal to eliminate complex fractions.
This process efficiently reduces the expression to the desired form, \(\sin x \cos x\).
Mastery of these steps aids in tackling even the most intricate of trigonometric identities.