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Mathematics
Mathematics
Problem
Let
f
:
R
∖
{
−
π
,
π
}
f:\mathbb{R}\setminus\{-\pi,\pi\}
f
:
R
∖
{
−
π
,
π
}
be a differentiable function such that
f
(
x
)
+
f
(
y
)
=
f
(
x
+
y
1
−
x
y
)
f(x) + f(y) = f\left(\frac{x+y}{1-xy}\right)
f
(
x
)
+
f
(
y
)
=
f
(
1
−
x
y
x
+
y
)
. If
f
(
1
)
=
π
f(1) = \pi
f
(
1
)
=
π
and
lim
x
→
0
f
(
x
)
=
2
\lim_{x \to 0} f(x) = 2
lim
x
→
0
f
(
x
)
=
2
. Then,
f
(
x
)
f(x)
f
(
x
)
is equal to a.
2
tan
−
1
x
2 \tan^{-1} x
2
tan
−
1
x
b.
1
2
tan
−
1
x
\frac{1}{2} \tan^{-1} x
2
1
tan
−
1
x
c.
π
2
tan
−
1
x
\frac{\pi}{2} \tan^{-1} x
2
π
tan
−
1
x
d.
2
π
tan
−
1
x
2\pi \tan^{-1} x
2
π
tan
−
1
x
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