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Question
Prove that:
`cos^(-1) 12/13 + sin^(-1) 3/5 = sin^(-1) 56/65`
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Solution
Let x = `cos^(-1) 12/13` 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 cos y = `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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