Verify that each equation is an identity.
tan² α sin² α = tan² α + cos² α - 1

Verified step by step guidance

Start by expressing \( \tan^2 \alpha \) in terms of \( \sin \alpha \) and \( \cos \alpha \). Recall that \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \), so \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \).

Substitute \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \) into the left side of the equation: \( \frac{\sin^2 \alpha}{\cos^2 \alpha} \sin^2 \alpha \).

Simplify the left side: \( \frac{\sin^4 \alpha}{\cos^2 \alpha} \).

Now, consider the right side of the equation: \( \tan^2 \alpha + \cos^2 \alpha - 1 \). Substitute \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \) into the right side.

Simplify the right side: \( \frac{\sin^2 \alpha}{\cos^2 \alpha} + \cos^2 \alpha - 1 \). Use the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \) to simplify further and verify if both sides are equal.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Common identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is crucial for verifying equations and simplifying trigonometric expressions.

Tangent and Sine Functions

The tangent function, defined as the ratio of sine to cosine (tan α = sin α / cos α), plays a key role in trigonometry. The sine function represents the ratio of the opposite side to the hypotenuse in a right triangle. Recognizing how these functions relate to each other is essential for manipulating and verifying trigonometric equations.

Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²). This theorem underpins many trigonometric identities, particularly the relationship between sine, cosine, and tangent. It is fundamental for deriving and verifying identities involving these functions.

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