1.

Prove that: `tan^(-1){pi/4+1/2 cos^(-1)a/b}+tan{pi/4-1/2 cos^(-1)a/b}=(2b)/a`

Answer» Let `cos^(-1)a/b)=theta`. Then, `a/b=costheta`.
`therefore` LHS `=tan(pi/4+1/1theta}=tan{pi/4-1/2theta}`
`=(1+tan(theta/2))/(1-tan(theta/2))+(1-tan(theta/2))/(1+tan(theta/2))`
`=(1+tan(theta/2)^(2)+1-tan(theta/2)^(2))/(1-tan^(2)theta/2)`
`=2{(1+tan^(2)(theta/2))/(1-tan^(2)theta/2}=2/(costheta)=(2b)/a`=RHS
Hence, LHS=RHS.


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