Prove that: 2 tan^-1(√((a-b)/(a+b)) tan((θ)/(2)))=cos^-1((b+a cosθ)/(a+b cosθ))

Name: Problem 194, Chapter 2: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-04-08T18:04:03-08:00
Duration: 1 min 39 s
Channel: Rukhmani Jain
Description: Prove that: 2 tan^-1(√((a-b)/(a+b)) tan((θ)/(2)))=cos^-1((b+a cosθ)/(a+b cosθ))

step 1 We are given to prove that 210 inverse, the root of a minus b upon a plus b, 10 theta by 2 shoul

Prove that: 2 tan^-1(√((a-b)/(a+b))...

Key Concepts

Inverse Trigonometric Functions

These functions, such as arctan and arccos, serve as the inverses of the tangent and cosine functions respectively. Their properties and definitions are central to solving equations by 'undoing' the trigonometric operation. Understanding the relationship between these functions, including their domains and ranges, allows one to convert between different inverse trigonometric expressions as seen in the proof of the given identity.

Half-Angle Formulas

Half-angle formulas play a crucial role in converting expressions involving a full angle into a form that uses half of the angle. These formulas simplify complex trigonometric expressions by expressing sine and cosine in terms of the tangent of half the angle. This technique is particularly useful in proving identities that relate functions of the full angle to functions of the half angle.

Trigonometric Identities

A thorough understanding of trigonometric identities is essential for transforming and simplifying expressions. Identities such as the Pythagorean identity, angle addition formulas, and double-angle formulas provide the tools to rewrite complex expressions into simpler or alternative forms. In proving the given identity, these identities facilitate the conversion between different inverse trigonometric forms.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to reveal underlying patterns or identities. In the context of trigonometric identities, careful algebraic handling of fractions, radicals, and trigonometric expressions is necessary to equate expressions that initially appear different. This skill is critical when combining half-angle formulas with inverse function properties.

Domain and Range Considerations

When working with inverse trigonometric functions, it is important to consider the domains and ranges of these functions to ensure that the expressions are well-defined and the identities are valid. Verifying that the transformations respect these constraints is a key step in rigorously establishing an identity, since it ensures that both sides of an equation describe the same angle within an acceptable interval.

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