Problem 20 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity. $$\tan ^{2} y\left(\csc ^{2} y-1\right)=1$$

Step-by-Step Solution

Answer

The identity \(\tan ^{2} y(\csc ^{2} y - 1) = 1\) has been verified successfully.

Step 1: Write the Given Identity

The given identity is \(\tan ^{2} y(\csc ^{2} y - 1) = 1\)

Step 2: Substitute the Trigonometric Identities

Substitute \(\tan ^{2} y\) with \(\sin ^{2} y / \cos ^{2} y\), and \(\csc ^{2} y\) with \(1 / \sin ^{2} y\). Therefore, we have \(\tan ^{2} y(\csc ^{2} y - 1) = (\sin ^{2} y / \cos ^{2} y)((1 / \sin ^{2} y) - 1).\)

Step 3: Simplify the Expression

Simplify this expression to \((\sin ^{2} y / \cos ^{2} y)(1 - \sin ^{2} y)=1 - \sin ^{2} y\)

Step 4: Apply Pythagorean Identity

Using the Pythagorean identity \(\sin ^{2} y + \cos ^{2} y = 1\), substitute \(\cos ^{2} y\) with \(1 - \sin ^{2} y\). Therefore, we have \(1 - \sin ^{2} y = 1\)

Step 5: Final result

Equating the left-hand side of the equation to the right-hand side gives \(1 = 1\). This verifies the initial trigonometric identity.

Key Concepts

Tan Identity

The tangent (tan) identity is a fundamental aspect of trigonometry tied closely to the sine and cosine functions. The basic tan identity is expressed as \( \tan(y) = \frac{\sin(y)}{\cos(y)} \). This identity shows how tangent relates to sine and cosine, offering a way to express tangent in terms of other core trigonometric functions. In trigonometric problems, it's often more convenient to use sine and cosine directly. Thus, in the given problem to verify \( \tan^2 y(\csc^2 y - 1) = 1 \), we substitute \( \tan^2 y \) with \( \frac{\sin^2 y}{\cos^2 y} \).

This transformation allows us to simplify complex identities by expressing everything in terms of sine and cosine, which are often easier to manipulate and understand. Keep in mind that transformations and substitutions using the tan identity are invaluable when simplifying or verifying complex trigonometric equations.

Cosecant Identity

The cosecant identity is among the reciprocal identities in trigonometry. The cosecant function is defined as the reciprocal of the sine function, meaning \( \csc(y) = \frac{1}{\sin(y)} \). This identity can be extremely useful when dealing with trigonometric equations that involve both sine and cosecant, as it provides a bridge to express cosine purely in terms of sine.

In the problem, we utilize this identity by substituting \( \csc^2 y \) with \( \frac{1}{\sin^2 y} \). When we perform this substitution in our equation, \( \tan^2 y(\csc^2 y - 1) = \bigg(\frac{\sin^2 y}{\cos^2 y}\bigg)\bigg(\frac{1}{\sin^2 y} - 1\bigg) \) was formed, simplifying the equation considerably.

This simplification lets us work more directly with fractional identities and shows the importance of reciprocal identities in breaking down complex expressions into more manageable forms.

Pythagorean Identity

One of the most widely used concepts in trigonometry is the Pythagorean identity. The primary form of this identity is \( \sin^2 y + \cos^2 y = 1 \). It serves as a fundamental tool in establishing relationships between trigonometric functions. When dealing with problems involving squares of sine or cosine, the Pythagorean identity is often the go-to resource.

In verifying the identity, \( \tan^2 y(\csc^2 y - 1) = 1 \), we relied on the Pythagorean identity to replace \( 1 - \sin^2 y \) with \( \cos^2 y \). This substitution is vital as it provides a way to simplify expressions down to a level where they match or verify an identity.

Understanding this identity not only helps in these transformations but also aids in grasping the geometric implications of the circle, where \( \sin \) and \( \cos \) relate directly to the coordinates of a point on the unit circle. Mastery of the Pythagorean identity is crucial for successfully tackling trigonometric proofs and equations.