Question

what is the value of tan inverse minus 2 by 5​

Answer 1

Given :   Tan ⁻ ¹( - 2/5 )  

To Find :   Equivalent Value  in cot⁻¹  terms

Solution:

Principle Range of Tan⁻¹ is  ( -π/2 , π/2)

Tan ⁻ ¹( - 2/5 )   = -   Tan ⁻ ¹(2/5)

Principle Range of cot⁻¹ is  ( 0 , π )

Cot ⁻ ¹( - 5/2 )   =     π  -  cot ⁻ ¹(5/2)

Tan ⁻ ¹( x/y)  = cot⁻ ¹( y/x)    if   x and y are positive

=> Tan ⁻ ¹( 2/5)  = cot⁻ ¹( 5/2)    if   x and y are positive

⇒ Cot ⁻ ¹( - 5/2 )   =     π  -  Tan ⁻ ¹( 2/5)

=>  -  Tan ⁻ ¹( 2/5)  = Cot ⁻ ¹( - 5/2 )   -     π

Hence

Tan ⁻ ¹( - 2/5 )   = Cot ⁻ ¹( - 5/2 )   -     π

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