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Question
Find the value of the given expression.
`tan(sin^(-1) 3/5 + cot^(-1) 3/2)`
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Solution
`tan(sin^(-1) 3/5 + cot^(-1) 3/2)`
⇒ `sin^-1 3/5 = x`
⇒ `sin x = 3/5`
⇒ `cos x = sqrt(1 - sin^2 x) = 4/5`
⇒ `sec x = 5/4`
⇒ `tan x = sqrt(sec^2 - 1)`
⇒ `sqrt (25/16 - 1)`
⇒ `3/4`
⇒ `tan^-1 3/4`
⇒ `sin^-1 3/5 = tan^-1 3/4` ...(1)
⇒ `cot^-1 3/2 = tan^-1 2/3` ...(2)
⇒ `tan(sin^-1 3/5 + cot^-1 3/2)`
⇒ `tan (tan^-1 (((3/4 + 2/3))/(1 - 3/4 xx 2/3)))`
⇒ `tan (tan^-1 (((9 + 8)/12)/(1 - 1/2)))`
⇒ `tan(tan^-1 (17/6))`
= `17/6`
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