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Question
Find the value of the expression `sin(2tan^-1 1/3) + cos(tan^-1 2sqrt(2))`
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Solution
`sin(2tan^-1 1/3) + cos(tan^-1 2sqrt(2))`
⇒ `sin[tan^-1 ((2 xx 1/3)/(1 - (1/3)^2))] + cos[cos^-1 1/sqrt(1 + (2sqrt(2))^2)]` ......`[because tan^-1x = cos^-1 (1/sqrt(1 + x^2))]`
⇒ `sin[tan^-1 ((2/3)/(1 - 1/9))] + cos[cos^-1 (1/3)]`
⇒ `sin[tan^-1 (3/4)] + 1/3`
⇒ `sin[sin^-1 (3/5)] + 1/3`
⇒ `3/5 + 1/3`
⇒ `14/15` ......`[because tan^-1x = sin^-1 x/sqrt(1 + x^2)]`
Hence, `sin(2tan^-1 1/3) + cos(tan^-1 2sqrt(2)) = 14/15`
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