Question

The number of integral values of $x$ satisfying the equation ${\tan ^{ - 1}}\left( {3x} \right) + {\tan ^{ - 1}}\left( {5x} \right) = {\tan ^{ - 1}}\left( {7x} \right) + {\tan ^{ - 1}}\left( {2x} \right)$ is

Answer 1

Hint: Simplify the given equation using the formula, ${\tan ^{ - 1}}x + {\tan ^{ - 1}}y = {\tan ^{ - 1}}\left( {\dfrac{{x + y}}{{1 - xy}}} \right)$.

Complete step-by-step answer:
Now, simplify the equation by taking tan of the resultant equation on both sides, We will get an equation in $x$. Solve the equation to find the value of $x$. In the final answer, take only integral values of $x$.

We are given, ${\tan ^{ - 1}}\left( {3x} \right) + {\tan ^{ - 1}}\left( {5x} \right) = {\tan ^{ - 1}}\left( {7x} \right) + {\tan ^{ - 1}}\left( {2x} \right)$.
First let us simplify the given equations using the formula, ${\tan ^{ - 1}}x + {\tan ^{ - 1}}y = {\tan ^{ - 1}}\left( {\dfrac{{x + y}}{{1 - xy}}} \right)$.
$
\Rightarrow {\tan ^{ - 1}}\left( {3x} \right) + {\tan ^{ - 1}}\left( {5x} \right) = {\tan ^{ - 1}}\left( {7x} \right) + {\tan ^{ - 1}}\left( {2x} \right) \\
\Rightarrow {\tan ^{ - 1}}\left( {\dfrac{{3x + 5x}}{{1 - \left( {3x} \right)\left( {5x} \right)}}} \right) = {\tan ^{ - 1}}\left( {\dfrac{{7x + 2x}}{{1 - \left( {7x} \right)\left( {2x} \right)}}} \right) \\
\Rightarrow {\tan ^{ - 1}}\left( {\dfrac{{8x}}{{1 - 15{x^2}}}} \right) = {\tan ^{ - 1}}\left( {\dfrac{{9x}}{{1 - 14{x^2}}}} \right) \\
$
On taking tan on both sides, we get
\[\left( {\dfrac{{8x}}{{1 - 15{x^2}}}} \right) = \left( {\dfrac{{9x}}{{1 - 14{x^2}}}} \right)\]
Simplify the equation and solve for the value of $x$.
$
\Rightarrow 8x\left( {1 - 14{x^2}} \right) = 9x\left( {1 - 15{x^2}} \right) \\
\Rightarrow 8x - 112{x^3} = 9x - 135{x^3} \\
\Rightarrow 23{x^3} - x = 0 \\
\Rightarrow x\left( {23{x^2} - 1} \right) = 0 \\
\Rightarrow x = 0,x = \pm \dfrac{1}{{\sqrt {23} }} \\
$
The only integer value of $x$ is 0.
Hence, there is only one integral values of $x$satisfying the equation ${\tan ^{ - 1}}\left( {3x} \right) + {\tan ^{ - 1}}\left( {5x} \right) = {\tan ^{ - 1}}\left( {7x} \right) + {\tan ^{ - 1}}\left( {2x} \right)$.

Note: It is important to simplify question using the formula, ${\tan ^{ - 1}}x + {\tan ^{ - 1}}y = {\tan ^{ - 1}}\left( {\dfrac{{x + y}}{{1 - xy}}} \right)$ to avoid difficult calculations. After simplifying, we will consider only integral values of $x$.
In the final answer, we have to write the number of integral solutions and not the actual integral value of $x$, therefore, the answer will be 1 and not 0.