What-solutions-x-R-exist-for-the-equation-tan-1-x-cot-1-x-n-where-n-Z-

Question Number 5121 by Yozzii last updated on 15/Apr/16

W h a t s o l u t i o n s x \in R e x i s t f o r t h e

e q u a t i o n ({t a n}^{- 1} x) ({c o t}^{- 1} x) = n

w h e r e n \in Z ?

Answered by prakash jain last updated on 16/Apr/16

f (x) = \tan^{- 1} x \cot^{- 1} x

f^{'} (x) = \frac{\cot^{- 1} x - \tan^{- 1} x}{1 + x^{2}}

f^{'} (x) = 0 a t x = 1, x = - 1

f (x) m a x a t x = 1, x = - 1 and it takes value \frac{π^{2}}{16} .

The only possible value for n in question is 0 .

\tan^{- 1} x \cot^{- 1} x = 0 at x = 0 .

So x = 0 is the only solution .