Problem 59 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{1+\tan ^{2} u}{1-\tan ^{2} u}=\frac{1}{\cos ^{2} u-\sin ^{2} u} $$

Step-by-Step Solution

Answer

Identity verified; both sides simplify to \( \frac{1}{\cos^2 u - \sin^2 u} \).

Step 1: Recognize Trigonometric Identities

Recall the trigonometric identities: 1. The Pythagorean identity: \( an^2 u = rac{ an^2 u}{1} = rac{ an^2 u}{ ext{denominator}}\).2. The double angle identity for cosine: \( an^2 u = rac{ an^2 u + 1 - an^2 u}{2 an^2 u} == \).So , \( an^2 u + 1 - an^2 u = 1 \) and \( \cos^2 u - \sin^2 u = (2\tan^2 u -1) \)

Step 2: Substitute Identities into Left Side

Substitute the identity \( \tan^2 u = \frac{\sin^2 u}{\cos^2 u} \) into the left side expression:\[\frac{1 + \tan^2 u}{1 - \tan^2 u} = \frac{1 + \frac{\sin^2 u}{\cos^2 u}}{1 - \frac{\sin^2 u}{\cos^2 u}}\]

Step 3: Simplify the Fraction

Combine terms in the numerator and denominator:\[\frac{\cos^2 u + \sin^2 u}{\cos^2 u - \sin^2 u}.\]Recognize that \( \cos^2 u + \sin^2 u = 1 \), leading to:\[\frac{1}{\cos^2 u - \sin^2 u}\]

Step 4: Conclusion by Comparison

The simplified form of the left side \( \frac{1}{\cos^2 u - \sin^2 u} \) matches exactly with the right side of the equation. Thus, verifying the identity.

Key Concepts

Pythagorean identity

The Pythagorean identity is one of the cornerstone trigonometric identities. It states that for any angle \( u \), the sum of the squares of the sine and cosine of that angle equals one: \( \sin^2 u + \cos^2 u = 1 \). This relationship comes from the Pythagorean Theorem in a right triangle, where the hypotenuse is 1 on the unit circle. By rearranging this identity, you can derive other useful trigonometric identities:
\[ \sin^2 u = 1 - \cos^2 u \quad \text{and} \quad \cos^2 u = 1 - \sin^2 u \]
In the problem, the identity helps simplify expressions involving \( \tan^2 u \), which is linked to sine and cosine. The tangent squared identity, \( \tan^2 u = \frac{\sin^2 u}{\cos^2 u} \), further relies on this basic Pythagorean relationship to convert tangent terms into sine and cosine terms. Remembering the Pythagorean identity is a powerful tool for simplifying and verifying trigonometric expressions.

double angle identity

The double angle identities are critical for simplifying trigonometric expressions that involve angles that are double or triple. One particular double angle identity is \( \cos(2u) = \cos^2 u - \sin^2 u \). This identity allows expression of a double angle in terms of the square of sine and cosine.
This identity is particularly useful in verifying our original equation as it links the expression \( \cos^2 u - \sin^2 u \) directly with \( \cos(2u) \). Though the original problem may not visibly appear as a double angle expression, recognizing these connections can simplify both verification and transformation of complex trigonometric formulas.
Keep in mind other forms of the double angle for cosine, such as \( \cos(2u) = 2\cos^2 u - 1 \) and \( \cos(2u) = 1 - 2\sin^2 u \), which can be strategically used to simplify and restructure diverse problems involving angle doubling.

tan identity

The tangent identity can often serve as a bridge between sine and cosine due to its definition: \( \tan u = \frac{\sin u}{\cos u} \). In verifying trigonometric identities, recognizing \( \tan^2 u = \frac{\sin^2 u}{\cos^2 u} \) is essential. This view allows substitution and simplifying trigonometric expressions where tangent terms dominate.
The problem statement utilizes this identity by substituting \( \tan^2 u \) to realize its connection with the sine and cosine terms. This substitution transforms the original equation into a form aligned with well-known identities, enabling simplification.
Understanding and recognizing when to apply the tangent identity can dramatically streamline problem-solving processes in trigonometry. It's invaluable for converting expressions and verifying complex identities. Always remember: tangent ties back to sine and cosine, offering a means to switch between these terms smoothly.