Let $S = \left\{ {x \in R:0 < x < 1\,\mathrm{and}\,2{{\tan }^{ - 1}}\left( {{{1 - x} \over {1 + x}}} \right) = {{\cos }^{ - 1}}\left( {{{1 - {x^2}} \over {1 + {x^2}}}} \right)} \right\}$. If $\mathrm{n(S)}$ denotes the number of elements in $\mathrm{S}$ then :

Question

Let $S = \left\{ {x \in R:0 < x < 1\,\mathrm{and}\,2{{\tan }^{ - 1}}\left( {{{1 - x} \over {1 + x}}} \right) = {{\cos }^{ - 1}}\left( {{{1 - {x^2}} \over {1 + {x^2}}}} \right)} \right\}$.

If $\mathrm{n(S)}$ denotes the number of elements in $\mathrm{S}$ then :

$\mathrm{n}(\mathrm{S})=0$

$\mathrm{n}(\mathrm{S})=1$ and only one element in $\mathrm{S}$ is less than $\frac{1}{2}$.

$\mathrm{n}(\mathrm{S})=1$ and the elements in $\mathrm{S}$ is more than $\frac{1}{2}$.

$\mathrm{n}(\mathrm{S})=1$ and the element in $\mathrm{S}$ is less than $\frac{1}{2}$.

Solution

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