Prove the following for all y1, y2 ∈ℝ : (i) tan^-1 y1+tan^-1 y2=tan^-1((y1+y2)/(1-y1 y2)) if y1

step 1 In this question, we have to prove that 10 inverse x plus 10 inverse y equal to 1 upon 2 times s

Prove the following for all y1, y2 ∈ℝ : (i) tan^-1 y1+tan^-1...

Prove the following for all $y_{1}, y_{2} \in \mathbb{R}$ : (i) $\tan ^{-1} y_{1}+\tan ^{-1} y_{2}=\tan ^{-1}\left(\frac{y_{1}+y_{2}}{1-y_{1} y_{2}}\right)$ if $y_{1} y_{2}<1$, (ii) $\tan ^{-1}\left|y_{1}\right|+\tan ^{-1}\left|y_{2}\right|=\frac{\pi}{2}$ if $y_{1} y_{2}=1$ (iii) $\tan ^{-1}\left|y_{1}\right|+\tan ^{-1}\left|y_{2}\right|=\tan ^{-1}\left(\frac{\left|y_{1}\right|+\left|y_{2}\right|}{1-y_{1} y_{2}}\right)$ if $y_{1} y_{2}>1$.

Prove the following for all $y_{1}, y_{2} \in \mathbb{R}$ :
(i) $\tan ^{-1} y_{1}+\tan ^{-1} y_{2}=\tan ^{-1}\left(\frac{y_{1}+y_{2}}{1-y_{1} y_{2}}\right)$ if $y_{1} y_{2}<1$,
(ii) $\tan ^{-1}\left|y_{1}\right|+\tan ^{-1}\left|y_{2}\right|=\frac{\pi}{2}$ if $y_{1} y_{2}=1$
(iii) $\tan ^{-1}\left|y_{1}\right|+\tan ^{-1}\left|y_{2}\right|=\tan ^{-1}\left(\frac{\left|y_{1}\right|+\left|y_{2}\right|}{1-y_{1} y_{2}}\right)$ if $y_{1} y_{2}>1$.

Key Concepts

Inverse Tangent Function and its Properties

The inverse tangent, often denoted as tan?¹ or arctan, is the function that returns the angle whose tangent is a given number. Its principal value is typically taken in the interval (-?/2, ?/2), and understanding its range, domain, and continuity is vital when working with trigonometric equations that involve angle sums or transformations.

Tangent Addition Formula

The tangent addition formula, tan(? + ?) = (tan ? + tan ?)/(1 - tan ? tan ?), is a fundamental trigonometric identity that facilitates the computation of the tangent of a sum of angles. This formula underpins many proofs and derivations involving the inverse tangent function since applying it in reverse can provide a concise expression for the sum of two arctan expressions.

Case Analysis Based on Argument Product

When proving identities involving arctan functions, particularly those expressed with sums or transformations, it is essential to perform a case analysis based on the product of the arguments. This analysis divides the domain into distinct regions (such as when the product is less than, equal to, or greater than one) which can alter the behavior of the function and the applicability of certain formulas, ensuring the proof accounts for all possibilities.

Handling Absolute Values in Trigonometric Equations

Incorporating absolute values in trigonometric equations, especially within inverse trigonometric functions, is a strategy to manage the directional or sign ambiguities inherent in trigonometric functions. Recognizing when to use absolute values helps in formulating correct expressions and understanding how the functions behave when dealing with non-negative quantities, particularly when these quantities lead to special cases such as sums equal to ?/2.

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