Find the value of tan^-1((a1 x-y)/(a1 y+x))+tan^-1((a2-a1)/(1+a1 a2)) +tan^-1((a3-a2)/(1+a3 a2))

step 1 So we need to find the value of this expression. So I will be solving this term first. That is 1

Find the value of tan^-1((a1 x-y)/(a1...

Find the value of $\tan ^{-1}\left(\frac{a_{1} x-y}{a_{1} y+x}\right)+\tan ^{-1}\left(\frac{a_{2}-a_{1}}{1+a_{1} a_{2}}\right)$ $+\tan ^{-1}\left(\frac{a_{3}-a_{2}}{1+a_{3} a_{2}}\right)+\ldots .+\tan ^{-1}\left(\frac{a_{n}-a_{n-1}}{1+a_{n} a_{n-1}}\right)$ $+\tan ^{-1}\left(\frac{1}{a_{n}}\right)$, where $x, y, a_{1}, a_{2}, \ldots a_{n} \in R^{+}$

Find the value of $\tan ^{-1}\left(\frac{a_{1} x-y}{a_{1} y+x}\right)+\tan ^{-1}\left(\frac{a_{2}-a_{1}}{1+a_{1} a_{2}}\right)$
$+\tan ^{-1}\left(\frac{a_{3}-a_{2}}{1+a_{3} a_{2}}\right)+\ldots .+\tan ^{-1}\left(\frac{a_{n}-a_{n-1}}{1+a_{n} a_{n-1}}\right)$
$+\tan ^{-1}\left(\frac{1}{a_{n}}\right)$, where $x, y, a_{1}, a_{2}, \ldots a_{n} \in R^{+}$

Key Concepts

Inverse Trigonometric Functions

The inverse tangent function (tan?¹) gives an angle whose tangent is a specified number, effectively serving as the opposite of the tangent function. It is crucial in solving equations where the variable is wrapped inside a trigonometric function and plays a key role in converting between angle measures and ratio forms in various areas of mathematics.

Arctan Addition Formula

The arctan addition formula provides a method to combine two arctan expressions into a single arctan expression, given by tan?¹(u) + tan?¹(v) = tan?¹((u + v)/(1 - uv)) when the product uv is less than 1. This formula is indispensable in problems that require summing multiple inverse tangent terms and is a specific case of broader trigonometric addition formulas.

Telescoping Series

A telescoping series is a series in which successive terms cancel out part of the preceding terms, leading to a significant simplification of the sum. In the context of sums involving arctan differences, recognizing the telescoping pattern allows for reducing a complicated sum into a much simpler expression, often capturing only the first and last terms.

Trigonometric Identities

Trigonometric identities are equations that express relationships between trigonometric functions and are used to simplify or transform expressions. In problems that combine arctan functions, these identities enable the rewriting of terms in a form that makes the telescoping structure apparent, greatly simplifying the calculation.

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