Question
Question: The number of solutions for the equation \[{{\tan }^{-1}}\left( {{e}^{-x}} \right)+{{\cot }^{-1}}\le...
The number of solutions for the equation tan−1(e−x)+cot−1(∣lnx∣)=2π is:
(a) 0
(b) 1
(c) 3
(d) 2
Solution
We solve this problem first by taking the trigonometric equations out by using the standard formulas that is
tan−1x=cot−1(x1)
Then we use the composite angle formula of inverse trigonometric equations as
cot−1x+cot−1y=cot−1(x+yxy−1)
By using the above formulas we find the relation between to functions without trigonometric equations so that we can find the number of solutions using the graphs that is if f(x)=g(x) then the number of solutions to above equations will be number of points of intersections of y=f(x) and y=g(x)
Complete step-by-step solution
We are given that the equation that is
⇒tan−1(e−x)+cot−1(∣lnx∣)=2π
We know that the inverse trigonometric relation that is
tan−1x=cot−1(x1)
By using this relation to above equation we get