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Prove that 2sin^-1[3/5]-tan^-1[17/31]=pi...

Prove that `2sin^-1[3/5]-tan^-1[17/31]=pi/4`

Text Solution

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Let `sin^-1[3/5] = theta`,
Then, `sin theta = 3/5` `=>costheta = sqrt(1 - (3/5)^2) = 4/5`
`:. tan theta = sintheta/costheta = 3/4`
Now, `tan2theta = (2tantheta)/(1-tan^2theta) = (2**3/4)/(1-(3/4)^2) = (3/2)/(7/16) = 24/7`
`:. 2theta = tan^-1(24/7)`
Now, `L.H.S. = 2sin^-1(3/5) - tan^-1(17/31) = 2theta-tan^-1(17/31)`
`=tan^-1(24/7)-tan^-1(17/31)`
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