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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 1 - FunctionsProblema 87
Capitolo 1, Problema 87

Prove the following identities.
tan2θ=2tanθ1−tan2θ\(\tan\)2\(\theta\)=\(\frac{2\tan\theta}{1-\tan^2\theta}\)

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Start by recalling the double angle identity for tangent: \( \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \). This is a standard trigonometric identity.
To prove this identity, let's use the sine and cosine double angle identities: \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \) and \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \).
Express \( \tan(2\theta) \) in terms of sine and cosine: \( \tan(2\theta) = \frac{\sin(2\theta)}{\cos(2\theta)} \). Substitute the double angle formulas: \( \tan(2\theta) = \frac{2\sin(\theta)\cos(\theta)}{\cos^2(\theta) - \sin^2(\theta)} \).
Now, express \( \tan(\theta) \) as \( \frac{\sin(\theta)}{\cos(\theta)} \). Substitute this into the expression: \( \tan(2\theta) = \frac{2\left(\frac{\sin(\theta)}{\cos(\theta)}\right)}{1 - \left(\frac{\sin(\theta)}{\cos(\theta)}\right)^2} \).
Simplify the expression: \( \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \). This matches the given identity, thus proving it.

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Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are fundamental in simplifying expressions and solving equations in trigonometry. Common identities include the Pythagorean identities, reciprocal identities, and angle sum/difference identities, which provide relationships between different trigonometric functions.
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Verifying Trig Equations as Identities

Tangent Function

The tangent function, denoted as tan(θ), is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed in terms of sine and cosine as tan(θ) = sin(θ)/cos(θ). Understanding the properties and behavior of the tangent function is crucial for manipulating and proving identities involving tangent.
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Slopes of Tangent Lines

Double Angle Formulas

Double angle formulas are specific trigonometric identities that express trigonometric functions of double angles (2θ) in terms of single angles (θ). For example, the double angle formula for tangent is tan(2θ) = 2tan(θ)/(1 - tan²(θ)). These formulas are essential for simplifying expressions and proving identities, particularly in calculus and higher-level mathematics.
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Recursive Formulas
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