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Question
Prove that `sin^-1 3/5 - cos^-1 12/13 = sin^-1 16/65`
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Solution
`sin^-1 3/5 - cos^-1 12/13 = sin^-1 16/65`
x = `sin^-1 3/5`
son x = `3/5`
cos x = `4/5`
y = `cos^-1 2/13`
cos y = `12/13`
sin y = `5/13`
sin x = `3/5 = "opp"/"Hyp"`
Adj = `sqrt(5^2 - 3^2)`
= `sqrt(25 - 9)`
= `sqrt(16)`
= 4
cos y = `12/13 = "Adj"/"Hyp"`
Opp = `sqrt(13^2 - 12^2)`
= `sqrt(169 - 144)`
= `sqrt(25)`
= 5
`sin^-1 (3/5) - cos^-1 (12/13) = 3/5(12/13) - 4/5(5/13)`
`sin(x - y) = sinx cosy - cosx siny`
= `36/65 - 20/65`
= `sin(x - y) = 16/65`
`(x - y) = sin^-1(16/65)`
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