Question

solve:

cos-1 ([x2-1]/[x2+1]) + tan-1[(2x)/(x2-1)]=2pi/3

Answer 1

Concept Introduction:-

It is a type of branch in which mathematics  deals with their own properties and also manipulation of numbers.

Given:-

We have been given some numbers.

To Find:-

We have to find an answer.

Solution:-

According to the problem

cos-1 ( x2 – 1 / x2 + 1) + tan-1( 2x/ x2 – 1) = 2π/3

LHS = cos-1 ( x2 – 1 / x2 + 1) + tan-1( 2x/ x2 – 1)

Divided by x2 ,

cos-1 ( 1 – (1/x2) /( 1 + ( 1/x2) ) + tan-1( 2/x / 1 – (1/x2) )

= cos-1( 1 - (1/x)2 / 1 + (1/x)2 ) + tan-1 ( 2 (1/x) / 1 – (1/x)2 )

=2tan-1(1/x) + 2tan-1(1/x) [ using 2tan-1x = cos-1( 1 – x2 / 1 + x2 ) = tan-1( 2x / 1 – x2) ]

=4tan-1(1/x)

RHS = 2 π/3

4tan-1( 1/x ) = 2 π/3

tan-1(1/x) = π/6

1/x = 1/√3

x = √3

Final Answer:-

The final answer is √3

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