Problem 93 - Algebra and Trigonometry

Question

Verify each identity. \(\frac{\sin x-\cos x+1}{\sin x+\cos x-1}=\frac{\sin x+1}{\cos x}\)

Step-by-Step Solution

Answer

\(\frac{\sin x-\cos x+1}{\sin x+\cos x-1}\) simplifies to \(\frac{\sin x+1}{\cos x}\), therefore the trigonometric identity is verified.

Step 1: Rewrite the Left Hand Side

Start with the left-hand side (LHS) of the equation, i.e., \(\frac{\sin x-\cos x+1}{\sin x+\cos x-1}\). Rewrite this fraction by multiplying the numerator and denominator by \( \sin x + \cos x + 1 \). By doing this, a perfect square will form in the denominator. After doing the multiplication, we get: \(\frac{(\sin x-\cos x+1)(\sin x + \cos x + 1)}{(\sin x + \cos x - 1)(\sin x + \cos x + 1)}\)

Step 2: Simplify the Left Hand Side

Now, simplify the expressions. In the denominator, \((\sin x + \cos x + 1)\) cancels out. The numerator simplifies to the form of \((a-b)(a+b)= a^{2}- b^{2}\), where \(a= \sin x\) and \(b = \cos x\). This results in \(\sin^2 x - \cos^2 x + 2 \sin x\). In the denominator, use the same identity to get \(\sin^2 x - \cos^2 x\). Simplifying further, we get \(\frac{\sin^2 x - \cos^2 x + 2 \sin x}{\sin^2 x - \cos^2 x} = \frac{2 \sin x}{\sin^2 x - \cos^2 x} + 1\). But, \(\sin^2 x - \cos^2 x = -(1-\sin^2 x - \cos^2 x) = -\cos^2 x - \cos^2 x = -1\). Substituting this into our expression, we will get: \(\frac{\sin x+1}{\cos x}\). This expression is identical to the right-hand side (RHS).

Step 3: The Identity is Verified

By applying the identity \(\sin^2 x+ \cos^2 x = 1\) and simplifying the result, it has been shown that the original LHS is indeed equal to the RHS.

Key Concepts

Verifying Trigonometric Identities

Trigonometric identities play a critical role in simplifying expressions and solving trigonometry problems. Verifying an identity means to prove that the left-hand side (LHS) of an equation is equal to the right-hand side (RHS) without simply substituting values for the variable. This process requires a strong foundation in trigonometric properties and identities.

To verify an identity, one typically manipulates the LHS, the RHS, or both, using algebraic and trigonometric operations to reach the same expression on both sides of the equation. These operations may include factoring, distributing, combining like terms, and using fundamental identities like Pythagorean, reciprocal, or ratio identities.

Common Strategies

• Convert all expressions to sines and cosines.
• Factor and simplify using algebra.
• Use fundamental identities to substitute equivalent expressions.
• Combine fractions by finding common denominators.
• Cancel and reduce where possible.

For the exercise given, the problem-solving approach intuitively involved multiplying by a conjugate to create a perfect square, which is a clever strategy because it frequently simplifies trigonometric fractions. The verification was successful when both sides of the identity were simplified to the same expression.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is an essential skill that involves rewriting expressions to a more basic or more useful form. The goal could be to make the expression easier to work with, whether for integration, differentiation, or solving equations. Simplification is not just about making an expression shorter; it can also mean transforming it into a form that is more suitable for a particular application.

Tips for Simplification

• Understand and memorize key trigonometric identities.
• Look for patterns that match these identities.
• Break down complex expressions into simpler ones.
• Combine terms strategically to use identities effectively.
• Be patient and try different manipulations.

The provided exercise illustrates simplification wherein the numerator is expanded to apply the difference of squares, and the expression is then broken down further. The power of simplification is evident as the identity becomes clearer after each step. It is also crucial to recognize that simplification often involves working with fractions, which includes finding common denominators or canceling common factors.

Trigonometry Problem Solving

Trigonometry problem solving encompasses a wide range of techniques that are used to tackle problems involving angles, lengths, and the relationships between them. It is not only about knowing which formulas to apply, but also about understanding how to deconstruct a problem into manageable parts and then reconstruct the solution in a logical way.

Effective Problem-Solving Techniques

• Analyze the given problem and identify which trigonometric identities or formulas might be relevant.
• Draw diagrams where applicable to visualize the problem.
• Break the problem down into smaller, more manageable steps.
• Consider multiple angles of approach if a solution isn’t immediately apparent.
• Practice with various problems to build intuition and skill.

The step-by-step solution of the exercise demonstrates how to translate a verification problem into a series of algebraic and trigonometric steps. It shows the power of strategic manipulation and highlights the importance of perseverance. Problem-solving in trigonometry often requires trying several pathways before finding a route that leads to the simplified expression or the proof of an identity.