Prove the following identities.tan⁡2θ=2tan⁡θ1−tan⁡2θ\tan2\theta=\frac{2\tan\theta}{1-\$$tan^2$$\theta}tan2θ=1−tan2θ2tanθ

Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 1 - Functions

Problem 87

Chapter 1, Problem 87

Prove the following identities.
$\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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Key Concepts

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

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.

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.

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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Prove the following identities.tan2θ=2tanθ1−tan2θ\(\tan\)2\(\theta\)=\(\frac{2\tan\theta}{1-\tan^2\theta}\)tan2θ=1−tan2θ2tanθ

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Key Concepts

Trigonometric Identities

Tangent Function

Double Angle Formulas

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