Simplify
tan^ -1 [ x sqrt 9-x^ 2 ] Rightarrow ,|x|<3
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Answer:
Let x=tanθ, then θ=tan
−1
x
tan
−1
(
x
1+x
2
−1
)=tan
−1
(
tanθ
1+tan
2
θ
−1
)
=tan
−1
(
tanθ
sec
2
θ
−1
)
=tan
−1
(
tanθ
secθ−1
)
=tan
−1
[
cosθ
sinθ
cosθ
1
−1
]
=tan
−1
[
sinθ
1−cosθ
]
=tan
−1
[
2sin
2
θ
cos
2
θ
2sin
2
2
θ
]
=tan
−1
[
cos
2
θ
sin
2
θ
]
=tan
−1
[tan
2
θ
]
=
2
θ
=
2
tan
−1
x
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