Problem 43 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The identity is verified as both sides equal \( \sec^2 y \).

Step 1: Recognize trigonometric identity

First, recognize that \( 1 - \sin^2 y \) is the Pythagorean identity: \( \cos^2 y \). Rewrite the left-hand side of the equation to start with this identity.

Step 2: Rewrite left-hand side

Substitute \( \cos^2 y \) for \( 1 - \sin^2 y \) in the left-hand side fraction: \( \frac{1}{\cos^2 y} \).

Step 3: Recognize connection to secant

Recognize that \( \frac{1}{\cos^2 y} = \sec^2 y \). This relates the expression to the identity involving tangent.

Step 4: Use the Pythagorean identity for tangent and secant

Recall the Pythagorean identity \( \sec^2 y = 1 + \tan^2 y \). This identity shows the connection between secant and tangent.

Step 5: Confirm identity

By substituting \( \sec^2 y = 1 + \tan^2 y \), you verify that both sides of the equation are equal: \( \sec^2 y = 1 + \tan^2 y \). Therefore, the identity is confirmed to be true.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental trigonometric relation arising from the Pythagorean theorem. It expresses the relationship between the square of the sine and cosine of an angle. The most common form of this identity is given by the equation

• \( \sin^2 y + \cos^2 y = 1 \)

This identity indicates that the sum of the squares of the sine and cosine of any angle is always equal to one. It's a key tool for simplifying expressions in trigonometry.
Furthermore, by rearranging this equation, we derive two other useful trigonometric identities:

• \( \sin^2 y = 1 - \cos^2 y \)
• \( \cos^2 y = 1 - \sin^2 y \)

These forms are especially helpful when working with equations like the one in our exercise, where replacing \( 1 - \sin^2 y \) with \( \cos^2 y \) simplifies the problem significantly. Always remember, knowing the Pythagorean Identity can save you a lot of trouble in solving trigonometric equations.

Tangent Function

The tangent function, often denoted as \( \tan(y) \), is another important trigonometric function. It is fundamentally defined in terms of the sine and cosine functions. Specifically:

• \( \tan(y) = \frac{\sin(y)}{\cos(y)} \)

This relation explains why the tangent function is undefined for angles where the cosine is zero, like \( 90^\circ \) and \( 270^\circ \), as division by zero is not allowed.
In the context of identities, the tangent function is closely linked to our Pythagorean identity variations. We have:

• \( \sec^2 y = 1 + \tan^2 y \)

This relation is incredibly useful when transforming or verifying trigonometric identities, as seen in the given exercise. Recognizing these relationships helps in simplifying complex trigonometric expressions.
So, understanding the behavior and properties of the tangent function is crucial in mastering trigonometric concepts.

Secant Function

The secant function, represented as \( \sec(y) \), is one of the lesser-discussed trigonometric functions, but it plays a pivotal role in many trigonometric identities and problems. The secant is the reciprocal of the cosine function:

• \( \sec(y) = \frac{1}{\cos(y)} \)

Its significance becomes apparent when working with trigonometric identities involving tangent, especially those derived from Pythagorean identities.

• \( \sec^2 y = 1 + \tan^2 y \)

As shown in the solution above, when you replace \( \frac{1}{1 - \sin^2 y} \) with \( \sec^2 y \), you validate the identity using this secant-tangent relation.
Understanding the secant function as an extension of the reciprocal relationships among trig functions helps cement these connections, making problem-solving more intuitive. Always remember, in trigonometry, sometimes looking at the reciprocal functions can simplify your path to the solution!