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Prove that: tan^-1(1/4)+tan^-1(2/9)=1/2 ...

Prove that: `tan^-1(1/4)+tan^-1(2/9)=1/2 cos^-1(3/5)`.

Text Solution

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`tan^(-1)1/4+tan^(-1)2/9=tan^(-1)((1/4+2/9)/(1-1/4*2/9))`
`tan^(-1)((17/36)/(34/36))`
`tan^(-1)(1/2)`
`2tan^(-1)(1/2)=tan^(-1)(1/2)+tan^(-1)(1/2)`
`=tan^(-1)(1/(1-1/4))`
`=tan^(-1)(4/3)`
`tan^(-1)(1/4)+tan^(-1)(2/9)=1/2cos^(-1)(3/5)`
`tan^(-1)(1/2)=1/2cos^(-1)(3/5)`
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