Problem 103 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \(\sin ^{2} \alpha \sec ^{2} \alpha+\sin ^{2} \alpha \csc ^{2} \alpha=\sec ^{2} \alpha\)

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to \( \sec^2 \alpha \).

Step 1: Simplify the identity

Start by rewriting the given equation: \( \sin^2 \alpha \sec^2 \alpha + \sin^2 \alpha \csc^2 \alpha = \sec^2 \alpha \). We aim to simplify both sides of the equation to verify the identity.

Step 2: Express Reciprocal Trigonometric Functions

Rewrite \( \sec^2 \alpha \) and \( \csc^2 \alpha \) in terms of \( \sin \alpha \) and \( \cos \alpha \). We know that \( \sec^2 \alpha = \frac{1}{\cos^2 \alpha} \) and \( \csc^2 \alpha = \frac{1}{\sin^2 \alpha} \). Substitute these into the left side of the identity.

Step 3: Simplify the Left Side

Substitute the reciprocal expressions: \( \sin^2 \alpha \frac{1}{\cos^2 \alpha} + \sin^2 \alpha \frac{1}{\sin^2 \alpha} \). This simplifies to \( \frac{\sin^2 \alpha}{\cos^2 \alpha} + 1 \).

Step 4: Find a Common Denominator

The expression \( \frac{\sin^2 \alpha}{\cos^2 \alpha} + 1 \) requires a common denominator to combine the terms. Rewrite \( 1 \) as \( \frac{\cos^2 \alpha}{\cos^2 \alpha} \). The left side becomes \( \frac{\sin^2 \alpha + \cos^2 \alpha}{\cos^2 \alpha} \).

Step 5: Use the Pythagorean Identity

Apply the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \). Therefore, the expression \( \frac{\sin^2 \alpha + \cos^2 \alpha}{\cos^2 \alpha} \) simplifies to \( \frac{1}{\cos^2 \alpha} \), which is \( \sec^2 \alpha \).

Step 6: Verify Identity

Now both the left side and the right side of the equation simplify to \( \sec^2 \alpha \). Since they are equal, the identity \( \sin^2 \alpha \sec^2 \alpha + \sin^2 \alpha \csc^2 \alpha = \sec^2 \alpha \) is verified.

Key Concepts

Pythagorean Identity

When dealing with trigonometric identities, one of the most fundamental tools is the Pythagorean Identity. This identity states that for any angle \( \alpha \), the equation \( \sin^2 \alpha + \cos^2 \alpha = 1 \) holds true. This relation is intrinsic to the unit circle, where the square of the sine and cosine of an angle sum to one. This means that if you know one of these values, you can directly compute the other. For example:

• If \( \sin^2 \alpha = 0.4 \), then \( \cos^2 \alpha = 0.6 \).
• This can be solved as \( \cos \alpha = \sqrt{0.6} \) (consider the quadrant for sign), and vice versa for sine.

The Pythagorean Identity is often used to simplify or transform expressions in trigonometry, as it can bridge equations relating sine and cosine functions. In our exercise, this identity was crucial to simplify and verify the given trigonometric identity.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are inverses of the basic trigonometry functions, and they can be helpful in transforming and simplifying expressions. The three primary reciprocal functions are:

• Cosecant (\( \csc \alpha = \frac{1}{\sin \alpha} \))
• Secant (\( \sec \alpha = \frac{1}{\cos \alpha} \))
• Cotangent (\( \cot \alpha = \frac{1}{\tan \alpha} \))

Understanding and utilizing these reciprocal relationships is vital, as it allows the substitution of complex sine and cosine expressions with simpler fraction forms. In the provided exercise, rewriting \( \sec^2 \alpha \) and \( \csc^2 \alpha \) was a key step in verifying the identity. By expressing these terms in terms of \( \sin \alpha \) and \( \cos \alpha \) as \( \sec^2 \alpha = \frac{1}{\cos^2 \alpha} \) and \( \csc^2 \alpha = \frac{1}{\sin^2 \alpha} \), calculations become more straightforward.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is an essential skill in solving trigonometric identities and equations. The process generally involves:

• Using identities like the Pythagorean Identity: They help in reducing expressions to simpler forms.
• Applying reciprocal identities: They convert expressions into easier formats by fracturing them into sine, cosine, or tangent terms.
• Finding common denominators: This helps in combining fractions into single expressions, like what was done in our exercise, where we needed a common denominator to combine two terms.

The goal is to reduce complex trigonometric equations to basic forms which are easier to compare and verify. In our example, after substituting reciprocal functions and applying the Pythagorean Identity, we successfully simplified the left side of the equation to \( \sec^2 \alpha \), matching the right side and thus verifying the identity. Being able to see how different parts relate and reduce allows for better problem-solving and deeper understanding of trigonometric principles.