write the simplest form cos ^ (-1) x /√ x^2 +a^2
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Let x=tanθ⟹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
[
cosθ
1−cosθ
×
sinθ
cosθ
]
=tan
−1
⎣
⎢
⎢
⎡
2sin
2
θ
⋅cos
2
θ
2sin
2
2
θ
⎦
⎥
⎥
⎤
=tan
−1
(tan
2
θ
)=
2
θ
=
2
1
(tan
−1
x)
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