Question

what is tan(2tan-inverse(1/3))​

Answer 1

Ans ---> 3 / 4

To find ---> tan { 2tan⁻¹ ( 1 / 3 ) }

Solution---> We have two formulee as follows

(1) 2 tan⁻¹x = tan⁻¹ { 2x / ( 1 - x² ) }

(2) tan⁻¹ ( tanθ ) = θ

Now returning to original problem , we get

tan { 2 tan⁻¹ ( 1 / 3 ) }

Applying first formula from above , we get

= tan tan⁻¹ { 2 ( 1/3 ) / { 1 - ( 1 / 3 )² }

= tan tan⁻¹ [ ( 2 / 3 ) / { 1 - ( 1 / 9 ) } ]

= tan tan⁻¹ {( 2 / 3 ) / ( 9- 1 ) /9 }

= tan tan⁻¹ { ( 2 / 3 ) / (8 /9 )}

= tan tan⁻¹ ( 18 / 24 )

= tan tan⁻¹ ( 3 / 4 )

Applying second formula from above , we get

= 3 / 4

Additional information --->

(1) tan⁻¹ x + tan⁻¹ y = tan⁻¹ ( x + y / 1 - xy )

(2) tan⁻¹ x - tan⁻¹ y = tan⁻¹ ( x - y / 1 + xy )