How can you solve and equation with inverse functions?

Question

$$\arctan(x) - \arctan(2/x) = \arctan(7/9)$$ where $x$ is positive .

The answer should be 3. Thanks

Welcome to MSE! Please show your working, including where you are stuck. In addition, it is always better to format your questions with MathJax. — KM101, Jan 28 '19 at 17:28

score 2 · Accepted Answer · answered Jan 28 '19 at 17:29

2

Using $$\tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}$$ we get the equation $$\frac{x-2/x}{1+2}=\frac79$$ which looks like a nice quadratic equation.

answered Jan 28 '19 at 17:29

Angina Seng

158,341

I apologise for my lack of knowledge but could you elaborate on how you got that $\tan (A-B)=\frac 79, \tan A=x$ and $\tan B=\frac 2x$? – For the love of maths Jan 28 '19 at 17:33
Take the tangent of both sides. – KM101 Jan 28 '19 at 17:35
@KM101 thank you for the reply. – For the love of maths Jan 28 '19 at 17:38
Thank you sooo much! – Doris Vella Jan 28 '19 at 17:41

score 0 · Answer 2 · answered Jan 28 '19 at 17:59

Using Inverse trigonometric function identity doubt: $\tan^{-1}x+\tan^{-1}y =-\pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right)$, when $x<0$, $y<0$, and $xy>1$,

$$\arctan\dfrac2x+\arctan\dfrac79=\arctan\dfrac{18+7x}{9x-14}$$ for $\dfrac{2\cdot7}{x\cdot9}<1$ which happens if $x<0$ or if $x>\dfrac{14}9$

What if $\dfrac{14}{9x}\ge1?$

How can you solve and equation with inverse functions?

2 Answers2