Question

If $x < 0,y < 0$ such that $xy=1$ , then write the value of ${{\tan }^{-1}}x+{{\tan }^{-1}}y$ .

Answer 1

Hint: We will directly use the formula of ${{\tan }^{-1}}x+{{\tan }^{-1}}y$ which is given as ${{\tan }^{-1}}x+{{\tan }^{-1}}y={{\tan }^{-1}}\left( \dfrac{x+y}{1-xy} \right)$ . Thus, we will substitute the value of $xy=1$ and on solving we will get the answer. Further we will convert it in radian form by multiplying it with $\dfrac{\pi }{180{}^\circ }$ . Thus, we will get an answer.

Complete step-by-step answer:
We know that inverse function of tangent function i.e. $y={{\tan }^{-1}}\left( x \right)$ is in range of $-\dfrac{\pi }{2}$$<$$y$$<$$\dfrac{\pi }{2}$ for all real numbers.
Here, we will use the formula of ${{\tan }^{-1}}x+{{\tan }^{-1}}y$ as ${{\tan }^{-1}}x+{{\tan }^{-1}}y={{\tan }^{-1}}\left( \dfrac{x+y}{1-xy} \right)$ . On using this we get as
${{\tan }^{-1}}x+{{\tan }^{-1}}y={{\tan }^{-1}}\left( \dfrac{x+y}{1-xy} \right)$
Now, we will put value $xy=1$ in the above equation. We get as
${{\tan }^{-1}}x+{{\tan }^{-1}}y={{\tan }^{-1}}\left( \dfrac{x+y}{1-1} \right)$
${{\tan }^{-1}}x+{{\tan }^{-1}}y={{\tan }^{-1}}\left( \dfrac{x+y}{0} \right)$
Now, we know that if zero is in the denominator then value becomes infinity i.e. $\infty $ . So, we can write it as
${{\tan }^{-1}}x+{{\tan }^{-1}}y={{\tan }^{-1}}\left( \infty \right)$
Now, we know that $\tan 90{}^\circ =\infty $ . So, we write it as
${{\tan }^{-1}}x+{{\tan }^{-1}}y={{\tan }^{-1}}\left( \tan 90{}^\circ \right)$
Thus, we get as ${{\tan }^{-1}}x+{{\tan }^{-1}}y=90{}^\circ $ . In radian form, it is given as $90{}^\circ \times \dfrac{\pi }{180{}^\circ }=\dfrac{\pi }{2}$ .
Hence, the value of ${{\tan }^{-1}}x+{{\tan }^{-1}}y$ is $\dfrac{\pi }{2}$ .

Note:Remember that this formula is almost similar to $\tan \left( x+y \right)=\dfrac{\tan x+\tan y}{1-\tan x\tan y}$ . Just, here we have to find the inverse function value of the formula is little bit changed. Also, students should know that values of all angles i.e. $0{}^\circ ,30{}^\circ ,45{}^\circ ,60{}^\circ ,90{}^\circ $ so, that it becomes easy to solve. Do not make calculation mistakes.