Question

Let S ={x ∈ R :0<x <1 and 2 tan−1(1+x)/( 1− x ​)={cos−1(1-x 2)/(1+x 2​)}. If n(S) denotes the number of elements in S then :

• A. $n ( S )=2$ and only one element in $S$ is less than $\frac{1}{2}$.
• B. $n(S)=1$ and the element in $S$ is less than $\frac{1}{2}$.
• C. $n(S)=1$ and the elements in $S$ is more than $\frac{1}{2}$.
• D. $n(S)=0$

Answer 1

Answer: (B)

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

Answer 2

0<x<1
2tan−1(1+x1−x​)=cos−1(1+x21−x2​)
tan−1x=θ∈(0,4π​)
∴x=tanθ
2tan−1(tan(4π​−θ))=cos−1(cos2θ)
2(4π​−θ)=2θ
∴4θ=2π​∴θ=8π​
x=tan8π​
∴x=2​−1≃0.414