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Question

Show that 3tan-1x=tan-1(3x-x3)(1-3x2)


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Solution

To prove: 3tan-1x=tan-1(3x-x3)(1-3x2)

Let x=tanA

tan-1x=A

R.H.S=tan-1(3x-x3)(1-3x2)

=tan-1(3tanA-tanA3)(1-3tanA2)

=tan-1(tan3A) [Identity used: tan-1(3tanθ-tanθ3)(1-3tanθ2)]

= 3A

=3tan-1A

From above it is clear that L.H.S=R.H.S

Thus,3tan-1x=tan-1(3x-x3)(1-3x2)


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