Question

Prove that $2{{\tan }^{-1}}\dfrac{3}{4}-{{\tan }^{-1}}\dfrac{17}{31}=\dfrac{\pi }{4}$ .

Answer 1

Hint: The above question is related to inverse trigonometric function and for solving the problem, you need to use the formulas ${{\tan }^{-1}}A-{{\tan }^{-1}}B={{\tan }^{-1}}\dfrac{A-B}{1+AB}$ and ${{\tan }^{-1}}A+{{\tan }^{-1}}B={{\tan }^{-1}}\dfrac{A+B}{1-AB}$ .

Complete step-by-step answer:
Now moving to the solution to the above question, we will start with the left-hand side of the equation given in the question.
$2{{\tan }^{-1}}\dfrac{3}{4}-{{\tan }^{-1}}\dfrac{17}{31}$
$={{\tan }^{-1}}\dfrac{3}{4}+{{\tan }^{-1}}\dfrac{3}{4}-{{\tan }^{-1}}\dfrac{17}{31}$
Now, we know $\dfrac{3}{4}\times \dfrac{3}{4}<1$ . So, if we use the formula ${{\tan }^{-1}}A+{{\tan }^{-1}}B={{\tan }^{-1}}\dfrac{A+B}{1-AB}$ , we get
${{\tan }^{-1}}\dfrac{\dfrac{3}{4}+\dfrac{3}{4}}{1-\dfrac{3\times 3}{4\times 4}}-{{\tan }^{-1}}\dfrac{17}{31}$
$={{\tan }^{-1}}\dfrac{\dfrac{3}{2}}{\dfrac{16-9}{16}}-{{\tan }^{-1}}\dfrac{17}{31}$
$={{\tan }^{-1}}\dfrac{24}{7}-{{\tan }^{-1}}\dfrac{17}{31}$
Now we know that ${{\tan }^{-1}}A-{{\tan }^{-1}}B={{\tan }^{-1}}\dfrac{A-B}{1+AB}$ .
${{\tan }^{-1}}\dfrac{24}{7}-{{\tan }^{-1}}\dfrac{17}{31}={{\tan }^{-1}}\left( \dfrac{\dfrac{24}{7}-\dfrac{17}{31}}{1+\dfrac{24\times 17}{7\times 31}} \right)$
Now further solving the right-hand side of the above equation, we get
$={{\tan }^{-1}}\left( \dfrac{\dfrac{24\times 31-7\times 17}{7\times 31}}{\dfrac{7\times 13-24\times 17}{7\times 31}} \right)$
$={{\tan }^{-1}}\left( \dfrac{744-119}{217+408} \right)$
$={{\tan }^{-1}}\left( \dfrac{625}{625} \right)$
$={{\tan }^{-1}}1$
We know that the value of ${{\tan }^{-1}}1$ is equal to $\dfrac{\pi }{4}$ , which is equal to the right-hand side of the equation that we are asked to prove. So, we can say that we have proved that $2{{\tan }^{-1}}\dfrac{3}{4}-{{\tan }^{-1}}\dfrac{17}{31}=\dfrac{\pi }{4}$ .

Note: While dealing with inverse trigonometric functions, it is preferred to know about the domains and ranges of the different inverse trigonometric functions. For example: the domain of ${{\sin }^{-1}}x$ is $[-1,1]$ and the range is $\left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right]$ . Also it is important to check whether the multiplication of A and B is less than one or not to use proper formula.