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Question
Prove that `sin^(-1) (3/5) + cos^(-1) (12/13) = sin^(-1) (56/65)`
Solution
Let `cos^(-1) 12/13 = x`
∴ `cos x = 12/13`
∴ `sin x = 5/13`
and let `sin^(-1) 3/5 = y `
sin y = `3/5`
`:. cos y= 4/5`
∴ using sin (x + y) = sin x cos y + cos x sin y
`= 5/13xx4/5+ 12/13xx3/5`
`= (20+36)/(13xx5)`
= `56/65`
∴ x + y =`sin^(-1) 56/65`
`cos^(-1) 12/13 + sin^(-1) 3/5 = sin^(-1) 56/65`
Hence proved.
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