Question

Solution set of the equation $cos\{si{n}^{-1}\frac{1}{3}+co{s}^{-1}x\}=0$ has

• A. Infinitely many real numbers
• B. Only one real number
• C. Only three real numbers
• D. None of these

Answer 1

The correct option is B Only one real number

$cos\{si{n}^{-1}\frac{1}{3}+co{s}^{-1}x\}=0$
$\Rightarrow si{n}^{-1}\frac{1}{3}+co{s}^{-1}x=(2n-1)\frac{\pi }{2},nϵZ$
$\Rightarrow si{n}^{-1}x=(2n+1)\frac{\pi }{2}-si{n}^{-1}\frac{1}{3}$
$\Rightarrow si{n}^{-1}x=(2n+1)\frac{\pi }{2}-si{n}^{-1}\frac{1}{3}$
$\Rightarrow 0\le (2n+1)\frac{\pi }{2}-si{n}^{-1}\frac{1}{3}\le \pi ,nϵZ$
$\Rightarrow n=0$ $\Rightarrow co{s}^{-1}x=\frac{\pi }{2}-si{n}^{-1}\frac{1}{3}=co{s}^{-1}\frac{1}{3}$
$\Rightarrow x=\frac{1}{3}$
$\therefore$ Solution set has only one real number.