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Prove that `Sin^(-1) (3/5) + Cos^(-1) (12/13) = Sin^(-1) (56/65)` - Mathematics and Statistics

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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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APPEARS IN

 2017-2018 (March) (with solutions)
Question 3.2.1 | 4.00 marks
Solution Prove that `Sin^(-1) (3/5) + Cos^(-1) (12/13) = Sin^(-1) (56/65)` Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch.
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