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Question

If tan1(cosectan1xtancot1x)=atan1x(x0).
Find the value of a

A
a=1/2
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B
a=1
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C
a=2
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D
a=4
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Solution

The correct option is A a=1/2
We have, 2tan1(cosectan1xtancot1x)=2atan1x(x0)
Case (i):
x<0, put x=|x|
L.H.S. =2tan1(cosec(tan1(|x|))tan(cot1(|x|)))
=2tan1(cosec(tan1|x|)tan(πcot1|x|))

=2tan1(cosec(tan1|x|)+tan(cot1|x|))

=2tan1(cosec(cosec11+x2|x|)+tan(tan11|x|))

=2tan1(1+x2|x|+tan1|x|)
=2tan1(11+x2|x|)
=2tan1(11+tan2θtanθ)
=2tan1(1cosθtanθcosθ)
=2tan1(sin2θ/22sinθ/2cosθ/2)

=2tan1(tanθ2)=2×θ2=θ=tan1|x|
Case (ii):
x>0
L.H.S. =2tan1(cosec(tan1x)tan(cot1x))

=2tan1(cosec(cosec1(1+x2x))tan(tan11x))
=2tan1(1+x2x1x)
=2tan1(1+x21x)
=2tan1(secθ1tanθ)
=2tan1(cosθ+1sinθ)

=2tan1(2sinθ/22sinθ/2cosθ/2)=2tan1(tanθ2)=θ=tan1x
Hence a=1/2

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