Problem 45 - 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 since both sides equal \( \sec^2 y \).

Step 1: Recall Trigonometric Identities

We know the Pythagorean identities: \( \sin^2 y + \cos^2 y = 1 \) and \( 1 + \tan^2 y = \sec^2 y \). These identities will be useful in solving the given identity.

Step 2: Simplify the Left Side

Start with the left side \( \frac{1}{1 - \sin^2 y} \). Since \( 1 - \sin^2 y = \cos^2 y \), this simplifies to \( \frac{1}{\cos^2 y} \), which is equal to \( \sec^2 y \).

Step 3: Simplify the Right Side

Based on our knowledge of trigonometric identities, the right side \( 1 + \tan^2 y \) is equivalent to \( \sec^2 y \).

Step 4: Compare Both Sides

We find that both sides are equal: \( \sec^2 y = \sec^2 y \). Hence, the identity is verified.

Key Concepts

Pythagorean identities

One of the most essential components in understanding trigonometry is the Pythagorean identities. These are crucial tools used to simplify and verify trigonometric expressions in mathematics. The primary idea behind them is based on the Pythagorean Theorem, which connects the squares of the sides of a right triangle. In the world of trigonometry, especially when dealing with functions like sine, cosine, and tangent, these identities become incredibly handy.

The first Pythagorean identity, and perhaps the most famous one, is:

• \( \sin^2 y + \cos^2 y = 1 \)

This relation shows how the square of the sine function plus the square of the cosine function always equals one.

Another derived identity involves tangent and secant. By manipulating and substituting values into the basic identities, we arrive at another useful tool:

• \( 1 + \tan^2 y = \sec^2 y \)

Both identities are fundamental in calculus, physics, and various engineering fields. They not only help derive other identities but are also essential in simplifying complex trigonometric equations.

Simplifying trigonometric expressions

Simplifying trigonometric expressions may seem daunting, but with Pythagorean identities in hand, the task becomes straightforward. To simplify is to rewrite a complex expression in a more manageable form. In trigonometry, this often involves substituting Pythagorean identities or manipulating the expression to reveal a simpler equivalent.

For instance, in the exercise above, simplifying \( \frac{1}{1 - \sin^2 y} \) was a critical step. By recognizing that \( 1 - \sin^2 y = \cos^2 y \), we could substitute directly:

• \( \frac{1}{\cos^2 y} \) simplifies to \( \sec^2 y \)

This transformation is a clear illustration of how identities can simplify an expression to something more familiar and more accessible.

Whenever you encounter an unfamiliar trigonometric expression, consider the identities you know. They serve as a toolkit to break down and simplify expressions efficiently.

Verifying trigonometric identities

Verifying trigonometric identities involves proving that two different trigonometric expressions are indeed equal. This process requires a deep understanding of basic identities and an eye for strategic substitutions or transformations.

• Step one often includes simplifying either or both sides of the identity.
• Once simplification is complete, compare the expressions to verify equality.

In the provided exercise, we needed to show that \( \frac{1}{1 - \sin^2 y} = 1 + \tan^2 y \). By using the identities we know, both sides were simplified to \( \sec^2 y \), confirming their equality.

Remember, verifying is about establishing that no matter what angle you choose (in the appropriate domain), both expressions yield the same result, thus confirming the identity. Practicing this skill improves problem-solving capabilities, crucial for more advanced mathematics and scientific explorations.