Question

The equation arc cos x = arc tan x has

• A. Only one solution
• B. Exactly one solution in (0, 1)
• C. Exactly two solution in (-1, 1)
• D. No solution in (-1, 0)

Answer 1

Answer: (A), (B), (D)

The correct options are
A Only one solution
B Exactly one solution in (0, 1)
D No solution in (-1, 0)

As $co{s}^{-1}x\ge 0\forall xϵ[-1,1]$ and $ta{n}^{-1}x>0,\forall xin(0,\infty )$
So, solution must be a positive number only
Now, $co{s}^{-1}x=ta{n}^{-1}x$
$co{s}^{-1}x=co{s}^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right)\Rightarrow x^2=\frac{1}{1+x^2}$
$x^4+x^2-1=0\Rightarrow x^2=\frac{-1\pm \sqrt{5}}{2}\Rightarrow x=\sqrt{\frac{-1+\sqrt{5}}{2}}(0,1)$