Question

Prove that:

`cos^(-1)  12/13 + sin^(-1)  3/5 = sin^(-1)  56/65`

Answer 1

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)`