If |x|≤1,then 2tan⁻¹x -sin⁻¹(2x/(1+x²) =
Answers
Answer:
Dear Student,
Answer:Option a is correct. x= 1/√3
Solution:
1) First convert 2tan⁻¹x into cos⁻¹x, so that cos cancels cos⁻¹x
\begin{gathered}2 {tan}^{ - 1} x = {cos}^{ - 1} ( \frac{1 - {x}^{2} }{1 + {x}^{2} } ) \\ \\ \\ cos (\: 2 {tan}^{ - 1} x) = \\ \\ cos( {cos}^{ - 1} ( \frac{1 - {x}^{2} }{1 + {x}^{2} } ) = \frac{1}{2} \\ \\ ( \frac{1 - {x}^{2} }{1 + {x}^{2} } ) = \frac{1}{2} \\ \\ 2 - 2 {x}^{2} = 1 + {x}^{2} \\ \\ - 2 {x}^{2} - {x}^{2} = 1 - 2 \\ \\ - 3 {x}^{2} = - 1 \\ 3 {x}^{2} = 1 \\ {x}^{2} = \frac{1}{3} \\ \\ x = + - \sqrt{ \frac{1}{3} } \\ \\ x = - \frac{1}{ \sqrt{3} } \\ \\ x = - \frac{1}{ \sqrt{3} } \end{gathered}
2tan
−1
x=cos
−1
(
1+x
2
1−x
2
)
cos(2tan
−1
x)=
cos(cos
−1
(
1+x
2
1−x
2
)=
2
1
(
1+x
2
1−x
2
)=
2
1
2−2x
2
=1+x
2
−2x
2
−x
2
=1−2
−3x
2
=−1
3x
2
=1
x
2
=
3
1
x=+−
3
1
x=−
3
1
x=−
3
1
So, one of the value is 1/√3