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Question
Prove that:
`cos^(-1) 12/13 + sin^(-1) 3/5 = sin^(-1) 56/65`
Solution
Let x = `cos^-1 "and" y = sin^-1(3/5)`
or cos x` =12/13 "and" sin y = 3/5`
`sin x = sqrt (1 - cos^2 x) "and" cos y = sqrt(1 - sin^2 y)`
Now, `sin x = sqrt(1 - 144/169)` and `cosy = sqrt( 1 - 9/25)`
= `sin x = 5/13 "and" cos y = 4/5`
We know that,
sin (x + y) = sin x cos y + cos x sin y
= `5/13 xx 4/5 + 12/13 xx 3/5 `
= `20/65 + 36/65 `
= `56/65`
= `x + y = sin ^-1(56/65)`
or, `cos^-1(12/13) + sin^-1 (3/5)`
= `sin^-1(56/65)`
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