Verify that each equation is an identity.
sec² α - 1 = (sec 2α - 1)/(sec 2α + 1)

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Start by recalling the Pythagorean identity involving secant and tangent: \(\sec^{2} \alpha - 1 = \tan^{2} \alpha\).

Rewrite the left side of the equation using this identity: replace \(\sec^{2} \alpha - 1\) with \(\tan^{2} \alpha\).

Focus on simplifying the right side: \(\frac{\sec 2\alpha - 1}{\sec 2\alpha + 1}\). Express \(\sec 2\alpha\) in terms of cosine: \(\sec 2\alpha = \frac{1}{\cos 2\alpha}\).

Substitute \(\sec 2\alpha\) into the right side and simplify the complex fraction by multiplying numerator and denominator by \(\cos 2\alpha\) to clear denominators.

Use the double-angle identity for cosine, \(\cos 2\alpha = 1 - 2\sin^{2} \alpha\), and the Pythagorean identity \(\tan^{2} \alpha = \frac{\sin^{2} \alpha}{\cos^{2} \alpha}\) to rewrite and simplify the expression until both sides match.

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Verifying an identity means showing both sides simplify to the same expression, often using fundamental identities like Pythagorean or double-angle formulas.

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Fundamental Trigonometric Identities

Pythagorean Identity

The Pythagorean identity states that sin²α + cos²α = 1. From this, related identities like sec²α - 1 = tan²α are derived, which help transform and simplify expressions involving secant and tangent functions.

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Pythagorean Identities

Double-Angle Formulas

Double-angle formulas express trigonometric functions of 2α in terms of α, such as sec 2α = 1/cos 2α. These formulas are essential for rewriting and simplifying expressions involving sec 2α, enabling the verification of identities involving multiple angles.

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