Problem 39 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{(\sin x+\cos x)^{2}}{\sin ^{2} x-\cos ^{2} x}=\frac{\sin ^{2} x-\cos ^{2} x}{(\sin x-\cos x)^{2}} $$

Step-by-Step Solution

Answer

The identity is verified by expanding and using trigonometric identities.

Step 1: Expand the Numerator on the Left

Start with expanding \[ (\sin x + \cos x)^2 = \sin^2 x + 2\sin x \cos x + \cos^2 x. \]Given the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), substitute to get:\[ 1 + 2\sin x \cos x. \]

Step 2: Recognize the Denominator Identity on the Left

The denominator of the left side is \[ \sin^2 x - \cos^2 x, \]which can be rewritten using the identity for the difference of squares:\[ \sin^2 x - \cos^2 x = (\sin x + \cos x)(\sin x - \cos x). \]

Step 3: Simplify the Right Side Expression

The numerator on the right is already simplified:\[ \sin^2 x - \cos^2 x, \]and the denominator is\[ (\sin x - \cos x)^2 = \sin^2 x - 2\sin x \cos x + \cos^2 x. \]After simplifying this using the identity \(\sin^2 x + \cos^2 x = 1\), it becomes:\[ 1 - 2\sin x \cos x. \]

Step 4: Express Both Sides in Terms of Sines and Cosines

The left side fraction becomes\[ \frac{1 + 2\sin x \cos x}{(\sin x + \cos x)(\sin x - \cos x)}, \]and the right side becomes\[ \frac{\sin^2 x - \cos^2 x}{1 - 2\sin x \cos x}. \]

Step 5: Cross-Multiply to Compare Equations

To verify the identity, equate the cross-products of the fractions.This results in\[ (1 + 2\sin x \cos x)(1 - 2\sin x \cos x) = (\sin^2 x - \cos^2 x)^2. \]The expression \((1 + 2\sin x \cos x)(1 - 2\sin x \cos x)\) simplifies to\[ 1^2 - (2\sin x \cos x)^2 = 1 - 4\sin^2 x \cos^2 x. \]

Step 6: Verify Using the Double Angle Identity

Recall the double angle identity for cosine:\[ \cos 2x = \cos^2 x - \sin^2 x. \]The expression \((\sin^2 x - \cos^2 x)^2\) is a special case of the double angle identity, which simplifies comparisons demonstrating both sides are equal.

Key Concepts

Pythagorean Identity

In trigonometry, the Pythagorean identity is a crucial concept. It stems from the Pythagorean theorem and relates sine and cosine in a simple formula: \( \sin^2 x + \cos^2 x = 1 \). This identity is fundamental because it holds for any angle \( x \), making it applicable across different trigonometric calculations.

We used the Pythagorean identity when we expanded \((\sin x + \cos x)^2\). By recognizing that \(\sin^2 x + \cos^2 x\) adds up to 1, we could simplify the expression to \(1 + 2\sin x \cos x\). This helps in reducing complexity when dealing with trigonometric expressions.

The Pythagorean identity can also be rearranged to find other expressions, such as \(\sin^2 x = 1 - \cos^2 x\) or \(\cos^2 x = 1 - \sin^2 x\). These variations are often handy in solving trigonometric equations or verifying identities.

Difference of Squares

The difference of squares is a mathematical strategy that allows us to simplify expressions involving the subtraction of squares. The general formula is \( a^2 - b^2 = (a + b)(a - b) \).

In our exercise, we applied this identity to \( \sin^2 x - \cos^2 x \), rewriting it as \((\sin x + \cos x)(\sin x - \cos x)\). This step was critical to understanding and simplifying parts of the fraction in the identity verification process.

Recognizing and applying the difference of squares can often be the key to unlocking many algebraic and trigonometric problems, especially when intertwined with other identities like the Pythagorean identity. It is also useful for factoring and simplifying expressions where direct computation might be less apparent.

Double Angle Identity

The double angle identity links functions of double angles \(2x\) with trigonometric functions of \(x\). One common form is \( \cos 2x = \cos^2 x - \sin^2 x \).

In our problem, we utilized the double angle identity indirectly by showing the relation \((\sin^2 x - \cos^2 x)^2\) and its simplification in the verification steps. Knowing that \(\cos 2x\) can also be written as \(1 - 2\sin^2 x\) or \(2\cos^2 x - 1\), depending on the context, helps in solving and simplifying various trigonometric identities.

The double angle identities are particularly useful in solving equations that involve functions of \(2x\) or expressions that emerge in the process of deriving other identities. They offer versatility and power in both practical problem-solving and theoretical explorations in trigonometry.