Problem 77 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The identity \( \frac{\cos^2 t + \tan^2 t - 1}{\sin^2 t} = \tan^2 t \) is verified to be true.

Step 1: Understand the Trigonometric Identity

We need to verify that \( \frac{\cos ^{2} t+\tan ^{2} t-1}{\sin ^{2} t}=\tan ^{2}t \) is a true statement. The goal is to simplify the left-hand side of the equation to see if it matches the right-hand side.

Step 2: Apply Trigonometric Identities

Recall the identity \( \tan^2 t = \frac{\sin^2 t}{\cos^2 t} \). Replace \( \tan^2 t \) with \( \frac{\sin^2 t}{\cos^2 t} \) in the numerator of the left-hand side: \[\frac{\cos ^{2} t + \frac{\sin^2 t}{\cos^2 t} - 1}{\sin ^{2} t}.\]

Step 3: Simplify Using a Common Denominator

Combine the terms in the numerator over a common denominator, \( \cos^2 t \):\[\frac{\cos^{2} t + \frac{\sin^{2} t}{\cos^2 t} - 1}{\sin^{2} t} = \frac{\frac{\cos^{4} t + \sin^{2} t - \cos^{2} t}{\cos^2 t}}{\sin^{2} t}.\]

Step 4: Recognize Pythagorean Identity

Use the Pythagorean identity \( \sin^2 t + \cos^2 t = 1 \). Substitute \( \sin^2 t = 1 - \cos^2 t \) into the expression:\[\frac{\frac{\cos^{4} t + (1-\cos^2 t) - \cos^2 t}{\cos^2 t}}{\sin^{2} t}.\]

Step 5: Simplify the Numerator

Simplify the numerator by combining like terms:\[\cos^4 t + 1 - \cos^2 t - \cos^2 t = \cos^4 t - 2\cos^2 t + 1.\] Factor this expression as a perfect square:\[(\cos^2 t - 1)^2.\]

Step 6: Simplify the Fraction

Substitute back into the fraction and simplify:\[\frac{(\cos^2 t - 1)^2}{\cos^2 t \cdot \sin^{2} t}. \] Since \( \cos^2 t - 1 = -\sin^2 t \), we have:\[\frac{\sin^4 t}{\cos^2 t \cdot \sin^2 t} = \frac{\sin^2 t}{\cos^2 t}.\]

Step 7: Simplify to Final Expression

Recognize that the simplified form \( \frac{\sin^2 t}{\cos^2 t} \) is equal to \( \tan^2 t \). Thus, the left-hand side is equivalent to the right-hand side, verifying the identity.

Key Concepts

Pythagorean identity

In trigonometry, the Pythagorean identities are fundamental. They are named after the famous Pythagorean theorem, which deals with right triangles. The most commonly used Pythagorean identity is \( \sin^2 t + \cos^2 t = 1 \). This equation holds true for any angle \( t \), and it's a cornerstone in simplifying trigonometric expressions and solving equations.

To understand its usefulness in verifying identities, consider how it allows for substitution. For example, if you have \( \sin^2 t \), you can replace it with \( 1 - \cos^2 t \) in any given trigonometric expression. This becomes incredibly helpful when trying to manipulate an equation to show that one side equals the other, as in the example problem above.

Using this identity correctly can make complex equations easier to handle. It often reveals simpler forms or relationships between different trigonometric expressions. Remember, anytime you see \( \sin^2 t \) or \( \cos^2 t \), the Pythagorean identity is a tool you can use to transform the expression.

Simplifying expressions

Simplifying trigonometric expressions is similar to simplifying arithmetic expressions. The goal is to reduce the equation or expression to its simplest form to make calculations easier or to reveal underlying truths, like identities.

In trigonometry, simplification often involves using identities like the Pythagorean identity. Additionally, when working with fractions, finding a common denominator is essential. This was shown in the exercise solution where \( \tan^2 t \) was expressed as \( \frac{\sin^2 t}{\cos^2 t} \). This allowed us to combine terms in the numerator of a fraction.

Knowing how to factor expressions is another critical skill. For instance, the numerator \( \cos^4 t - 2\cos^2 t + 1 \) can be factored to \((\cos^2 t - 1)^2\). Recognizing these opportunities to factor expressions helps in simplifying complex trigonometric identities.

Verifying identities

Verifying trigonometric identities involves proving that two sides of an equation are equal, typically using various trigonometric properties and identities. This process requires converting one side of the equation, usually the more complex one, to match the other side.

In the provided exercise, the goal was to prove that \( \frac{\cos^2 t + \tan^2 t - 1}{\sin^2 t} = \tan^2 t \) is true. We did this by manipulating the left-hand side step by step. This required transforming \( \tan^2 t \) in terms of \( \sin^2 t \) and \( \cos^2 t \), and simplifying using the Pythagorean identity.

Verification becomes easier when we spot opportunities to substitute identities and simplify. Factoring, as seen in the example with \( (\cos^2 t - 1)^2 \), often helps simplify complex expressions to perfect squares or other simpler forms. This is essential in verifying identities because it makes the mechanical work of simplification more accessible and mistakes less likely.