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Question
Prove that:
`tan^(-1) 63/16 = sin^(-1) 5/13 + cos^(-1) 3/5`
Solution
Let `sin^(-1) 5/13 = x`. Then, `sin x = 5/13 `
=> `cos x = 12/13`
`:. tan x = 5/12 `
=> `x = tan^(-1) 5/12`
`:. sin^(-1) 5/13 `
`=tan^(-1) 5/12` ...(1)
Let `cos^(-1) 3/5 = y`. Then `cos y = 3/5 `
`=> sin y = 4/5`
`:. tan y = 4/3 `
`=> y = tan^(-1) 4/3`
`:. cos^(-1) 3/5 = tan^(-1) 4/3` .....(2)
Using (1) and (2), we have
R.H.S = `sin^(-1) 5/13 + cos^(-1) 3/5`
`= tan^(-1) 5/12 + cos^(-1) 4/5`
`=tan^(-1) 5/12 + tan^(-1) 4/3`
`= tan^(-1) ((5/12+ 4/3)/(1-5/12 xx 4/3))` `[tan^(-1)x + tan^(-1) y = tan^(-1) (x+y)/(1-xy)]`
`= tan^(-1) ((15+48)/(36-20))`
`= tan^(-1) 63/16`
= L.H.S
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