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Given an arithmetic progression $a_1, a_2, a_3, ..., a_n$ with a common difference d, what is the simplified form of the following expression?
$\tan^{-1}\left(\frac{d}{1 + a_1a_2}\right) + \tan^{-1}\left(\frac{d}{1 + a_2a_3}\right) + ... + \tan^{-1}\left(\frac{d}{1 + a_{n-1}a_n}\right)$
Given an arithmetic progression $a_1, a_2, a_3, ..., a_n$ with a common difference d, what is the simplified form of the following expression?
$\tan^{-1}\left(\frac{d}{1 + a_1a_2}\right) + \tan^{-1}\left(\frac{d}{1 + a_2a_3}\right) + ... + \tan^{-1}\left(\frac{d}{1 + a_{n-1}a_n}\right)$
- $\frac{(n-1)d}{1 + a_1a_n}$ (correct)
- $\frac{nd}{1 + a_1a_n}$
- $\frac{(n-1)d}{a_1 + a_n}$
- $\frac{a_n - a_1}{a_n + a_1}$
If $\tan^{-1}y = \tan^{-1}x + \tan^{-1}\left(\frac{2x}{1 - x^2}\right)$ and $|x| < \frac{1}{\sqrt{3}}$, determine the value of y.
If $\tan^{-1}y = \tan^{-1}x + \tan^{-1}\left(\frac{2x}{1 - x^2}\right)$ and $|x| < \frac{1}{\sqrt{3}}$, determine the value of y.
- $\frac{3x + x^3}{1 - 3x^2}$ (correct)
- $\frac{3x - x^3}{1 - 3x^2}$
- $\frac{3x + x^3}{1 + 3x^2}$
- $\frac{3x - x^3}{1 + 3x^2}$
Evaluate the expression: $\tan\left[2\tan^{-1}\left(\frac{1}{5}\right) - \frac{\pi}{4}\right]$
Evaluate the expression: $\tan\left[2\tan^{-1}\left(\frac{1}{5}\right) - \frac{\pi}{4}\right]$
- $-\frac{7}{17}$ (correct)
- $\frac{17}{7}$
- $\frac{7}{17}$
- $-\frac{17}{7}$
Determine the value of $\sin\left[3\sin^{-1}\left(\frac{1}{5}\right)\right]$.
Determine the value of $\sin\left[3\sin^{-1}\left(\frac{1}{5}\right)\right]$.
Find the value of $\cot\left[\cos^{-1}\left(\frac{7}{25}\right)\right]$.
Find the value of $\cot\left[\cos^{-1}\left(\frac{7}{25}\right)\right]$.
If $\cos(2\sin^{-1}x) = \frac{1}{9}$, what are the possible values of x?
If $\cos(2\sin^{-1}x) = \frac{1}{9}$, what are the possible values of x?
Given that $a_1, a_2, a_3, ..., a_n$ form an arithmetic progression with common difference d, how can $\tan\left[\tan^{-1}\left(\frac{d}{1 + a_1a_2}\right) + \tan^{-1}\left(\frac{d}{1 + a_2a_3}\right) + ... + \tan^{-1}\left(\frac{d}{1 + a_{n-1}a_n}\right)\right]$ be expressed?
Given that $a_1, a_2, a_3, ..., a_n$ form an arithmetic progression with common difference d, how can $\tan\left[\tan^{-1}\left(\frac{d}{1 + a_1a_2}\right) + \tan^{-1}\left(\frac{d}{1 + a_2a_3}\right) + ... + \tan^{-1}\left(\frac{d}{1 + a_{n-1}a_n}\right)\right]$ be expressed?
If $\tan^{-1} y = \tan^{-1} x + \tan^{-1} \left(\frac{2x}{1 - x^2}\right)$ and $|x| < \frac{1}{\sqrt{3}}$, which of the following is a valid expression for $y$?
If $\tan^{-1} y = \tan^{-1} x + \tan^{-1} \left(\frac{2x}{1 - x^2}\right)$ and $|x| < \frac{1}{\sqrt{3}}$, which of the following is a valid expression for $y$?
What is the exact value of $\tan\left(2 \arctan\left(\frac{1}{5}\right) - \frac{\pi}{4}\right)$?
What is the exact value of $\tan\left(2 \arctan\left(\frac{1}{5}\right) - \frac{\pi}{4}\right)$?
Given $\sin\left[3\sin^{-1}\left(\frac{1}{5}\right)\right]$, determine the value of the expression.
Given $\sin\left[3\sin^{-1}\left(\frac{1}{5}\right)\right]$, determine the value of the expression.
Determine the value of $\cot(\cos^{-1}(\frac{7}{25}))$.
Determine the value of $\cot(\cos^{-1}(\frac{7}{25}))$.
Given that $\cos(2\sin^{-1}(x)) = \frac{1}{9}$, find all possible values of x.
Given that $\cos(2\sin^{-1}(x)) = \frac{1}{9}$, find all possible values of x.
Consider an arithmetic progression $a_1, a_2, a_3, \dots, a_n$ with a common difference d. Which of the following expressions is equivalent to $\tan^{-1}\left(\frac{d}{1 + a_1 a_2}\right) + \tan^{-1}\left(\frac{d}{1 + a_2 a_3}\right) + \dots + \tan^{-1}\left(\frac{d}{1 + a_{n-1} a_n}\right)$?
Consider an arithmetic progression $a_1, a_2, a_3, \dots, a_n$ with a common difference d. Which of the following expressions is equivalent to $\tan^{-1}\left(\frac{d}{1 + a_1 a_2}\right) + \tan^{-1}\left(\frac{d}{1 + a_2 a_3}\right) + \dots + \tan^{-1}\left(\frac{d}{1 + a_{n-1} a_n}\right)$?
Let $\tan^{-1} y = \tan^{-1} x + \tan^{-1}\left(\frac{2x}{1 - x^2}\right)$, where $|x| < \frac{1}{\sqrt{3}}$. What is the value of $\tan y$?
Let $\tan^{-1} y = \tan^{-1} x + \tan^{-1}\left(\frac{2x}{1 - x^2}\right)$, where $|x| < \frac{1}{\sqrt{3}}$. What is the value of $\tan y$?
Calculate the exact value of $\tan\left[2\tan^{-1}\left(\frac{1}{5}\right) - \frac{\pi}{4}\right]$.
Calculate the exact value of $\tan\left[2\tan^{-1}\left(\frac{1}{5}\right) - \frac{\pi}{4}\right]$.
What is the value of $\sin\left[3\sin^{-1}\left(\frac{1}{5}\right)\right]$?
What is the value of $\sin\left[3\sin^{-1}\left(\frac{1}{5}\right)\right]$?
Determine the value of $\cot\left[\cos^{-1}\left(\frac{7}{25}\right)\right]$.
Determine the value of $\cot\left[\cos^{-1}\left(\frac{7}{25}\right)\right]$.
Find all the possible values of $x$ given the equation $\cos(2\sin^{-1}x) = \frac{1}{9}$.
Find all the possible values of $x$ given the equation $\cos(2\sin^{-1}x) = \frac{1}{9}$.
Flashcards
A.P. Series Sum with tan⁻¹
A.P. Series Sum with tan⁻¹
If a₁, a₂, a₃, ..., aₙ is an A.P. with common difference d, the sum of the series tan⁻¹[d/(1+a₁a₂)] + tan⁻¹[d/(1+a₂a₃)] + ... + tan⁻¹[d/(1+aₙ₋₁aₙ)] equals (n-1)d / (a₁ + aₙ).
tan⁻¹ Addition Formula
tan⁻¹ Addition Formula
If tan⁻¹y = tan⁻¹x + tan⁻¹(2x/(1-x²)), then y = (3x - x³) / (1 - 3x²) where |x| < 1/√3.
tan(2tan⁻¹(1/5) - π/4)
tan(2tan⁻¹(1/5) - π/4)
The value of tan[2tan⁻¹(1/5) - π/4] is -7/17
sin(3sin⁻¹(1/5)) Value
sin(3sin⁻¹(1/5)) Value
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cos(2sin⁻¹x) = 1/9, find x
cos(2sin⁻¹x) = 1/9, find x
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Study Notes
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If a₁, a₂, a₃, ..., aₙ is an arithmetic progression (A.P.) with common difference d, then:
- tan[tan⁻¹(d / (1 + a₁a₂)) + tan⁻¹(d / (1 + a₂a₃)) + ... + tan⁻¹(d / (1 + aₙ₋₁aₙ))] = (aₙ - a₁) / (1 + a₁aₙ)
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If tan⁻¹y = tan⁻¹x + tan⁻¹(2x / (1 - x²)), where |x| < 1/√3, then a value of y is (3x - x³) / (1 - 3x²)
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tan[2tan⁻¹(1/5) - π/4] = -7/17
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sin[3sin⁻¹(1/5)] = 74/125
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cot[cos⁻¹(7/25)] = 24/7
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If cos(2sin⁻¹x) = 1/9, then x = ±2/3
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