Question

$co{s}^{-1}x$ is equal to

• A. $2si{n}^{-1}\sqrt{\frac{1-x}{2}}$
• B. $2co{s}^{-1}\sqrt{\frac{1-x}{2}}$
• C. $2co{s}^{-1}\sqrt{\frac{1+x}{2}}$
• D. $2si{n}^{-1}\sqrt{\frac{1+x}{2}}$

Answer 1

Answer: (A), (C)

The correct options are
A $2si{n}^{-1}\sqrt{\frac{1-x}{2}}$
C $2co{s}^{-1}\sqrt{\frac{1+x}{2}}$

Let $co{s}^{-1}x=y$, so that $x=cosy$. Then

$\Rightarrow \sqrt{\frac{1-x}{2}}=\sqrt{\frac{1-cosy}{2}}=\sqrt{\frac{2si{n}^2\left(\frac{y}{2}\right)}{2}}=sin\frac{y}{2}$

$\Rightarrow si{n}^{-1}\sqrt{\frac{1-x}{2}}=\frac{y}{2}\Rightarrow 2si{n}^{-1}\sqrt{\frac{1-x}{2}}=y$

Also $\sqrt{\frac{1+x}{2}}=\sqrt{\frac{1+cosy}{2}}=\sqrt{\frac{2co{s}^2\left(\frac{y}{2}\right)}{2}}=cos\frac{y}{2}$

$\Rightarrow co{s}^{-1}\sqrt{\frac{1+x}{2}}=\frac{y}{2}\Rightarrow 2co{s}^{-1}\sqrt{\frac{1+x}{2}}=y$

$\therefore y=co{s}^{-1}x=2si{n}^{-1}\sqrt{\frac{1-x}{2}}=2co{s}^{-1}\sqrt{\frac{1+x}{2}}$