Name: Multiple Angle Formula for Sine Inverse Function
Uploaded: 2019-07-19
Duration: 6 min 56 s
Description: What is twice of sine inverse function? Can we develop a formula for it? How to solve the problem using this function? Check out this video to know more.

Question

(i)

Find all positive integral solutions of the equation,

\tan^{- 1} x + \cot^{- 1} y = \tan^{- 1} 3

(ii)

If

' k'

be a positive integer, then show that the equation:

\tan^{- 1} x + \tan^{- 1} y = \tan^{- 1} k

has no non-zero integral solution.

Accepted Answer

$(i) \tan^{- 1} x + \cot^{- 1} y = \tan^{- 1} 3$

$\Rightarrow \tan^{- 1} x + \frac{π}{2} - \tan^{- 1} y = \tan^{- 1} 3$

$\Rightarrow \tan^{- 1} x - \tan^{- 1} y = \tan^{- 1} 3 - \frac{π}{2}$

$\tan^{- 1} (\frac{x - y}{1 + x y}) = - (\frac{π}{2} - \tan^{- 1} 3)$

$\Rightarrow \frac{x - y}{1 + x y} = - \tan (\frac{π}{2} - \tan^{- 1} 3)$

$' k'$

$\Rightarrow \frac{x - y}{1 + x y} = - \frac{1}{3}$

If $y - x = λ, 1 + x y = 3 λ . . . (i)$

$1 + x (x + λ) = 3 λ$

$λ = \frac{x^{2} + 1}{3 - x}$

As $x, y > 0 & λ \in I$

As $x y > 0 \Rightarrow 3 λ - 1 > 0$ (from $(i)$ )

$λ > \frac{1}{3}$

For $x = 1, λ = 1, y = 2$

For $x = 2, λ = 5, y = 7$

For $x > 3, λ < 0$

$∴ 2$ solutions $(1, 2) & (2, 7)$ .

$(ii)$ Given, $\tan^{- 1} x + \tan^{- 1} y = \tan^{- 1} k$ , where $k$ is a positive integer.

We have to prove that $x$ and $y$ can not be integers.

Let us assume that $x$ and $y$ are integers

So, $\tan^{- 1} x + \tan^{- 1} y = \tan^{- 1} k$

Using the formula, $\tan^{- 1} x + \tan^{- 1} y = \tan^{- 1} (\frac{x + y}{1 - x y})$

$\Rightarrow \tan^{- 1} (\frac{x + y}{1 - x y}) = \tan^{- 1} k$

$\Rightarrow \frac{x + y}{1 - x y} = k$

As $x$ and $y$ are integers

$\Rightarrow x y \geq 1 \Rightarrow 1 - x y \leq 0$

$∴ \frac{x + y}{1 - x y}$ is negative $\Rightarrow k$ is negative

But this is contradictory, since it is given that $k$ is a positive integer.

Hence, $x$ and $y$ cannot be integers.

Hence proved.

$(i)$ Find all positive integral solutions of the equation, $\tan^{- 1} x + \cot^{- 1} y = \tan^{- 1} 3$
$(ii)$ If $' k'$ be a positive integer, then show that the equation:
$\tan^{- 1} x + \tan^{- 1} y = \tan^{- 1} k$ has no non-zero integral solution.

Important Questions on Inverse Trigonometric Functions

Learn and Prepare for any exam you want !

(i) Find all positive integral solutions of the equation, tan-1x+cot-1y=tan-13 (ii) If 'k' be a positive integer, then show that the equation: tan-1x+tan-1y=tan-1k has no non-zero integral solution.

Important Questions on Inverse Trigonometric Functions

Learn and Prepare for any exam you want !

$(i)$ Find all positive integral solutions of the equation, $\tan^{- 1} x + \cot^{- 1} y = \tan^{- 1} 3$
$(ii)$ If $' k'$ be a positive integer, then show that the equation:
$\tan^{- 1} x + \tan^{- 1} y = \tan^{- 1} k$ has no non-zero integral solution.