1.

Prove that `ta n^-1 1/3 + ta n ^-1 1/5 + ta n ^-1 1/7 + ta n ^-1 1/8 = pi/4`

Answer» We have,
LHS `=(tan^(-1)1/3+tan^(-1)1/5)+(tan^(-1)1/7+tan^(-1)1/8)`.
`=tan^(-1)((1/3 +1/5))/(1-1/3 xx 1/5) + tan^(-1)((1/7+1/8)/(1-1/7 xx 1/8))`
`=tan^(-1)((8/15)/(14/15)) + tan^(-1)((15/56)/(55/56)`
`=tan^(-1)8/14+tan^(-1)15/55=tan^(-1)4/7+tan^(-1)3/11`
`=tan^(-1)((4/7+3/11)/(1-4/7 xx 3/11))= tan^(-1)((65/77)/(65/77)) = tan^(-1)1=pi/4`= RHS.
`therefore` LHS=RHS.


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