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Cos(A+B)Cos(A-B)

If x takes ...

Question

If $x$ takes negative permissible value, then ${sin}^{- 1} x =$

{cos}^{- 1} \sqrt{1 - x^{2}}

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- {cos}^{- 1} \sqrt{1 - x^{2}}

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{cos}^{- 1} \sqrt{x^{2} - 1}

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π - {cos}^{- 1} \sqrt{1 - x^{2}}

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Solution

The correct option is B $- {cos}^{- 1} \sqrt{1 - x^{2}}$
If $- 1 \leq x < 0$ , then $\frac{- π}{2} \leq s i n^{- 1} (x) < 0$
Let, $s i n θ = x$
Hence, $\frac{- π}{2} \leq θ < 0$
Hence, $c o s θ = \sqrt{1 - x^{2}}$
Since, the range of $c o s^{- 1} (x)$ is $[0, π]$ ,
$θ = - c o s^{- 1} (\sqrt{1 - x^{2}})$

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