Question

Csc (2sin^-1(-12/13))
that is 2 times inverse sin of -12/13

Answer 1

Answer:
`-(169)/(120)`

I'm assuming you mean this:


`cosec(2sin^(-1)((-12)/(13)))`

Recall what the cosecant function represents. It is the reciprocal of a sine function, and is often written in this form:


`(1)/(sinx)`

So, using the sine relationship:


`cosec(2sin^(-1)((-12)/(13))) = (1)/(sin(2sin^(-1)((-12)/(13))))`

Let
`u = sin^(-1)((-12)/(13))`

`cosec(2sin^(-1)((-12)/(13))) = (1)/(sin(2u))`

`= (1)/(2sinu \cdot cosu)`

`= (1)/(2sin(sin^(-1)((-12)/(13))) \cdot cos(sin^(-1)((-12)/(13))))`

`= \frac{1}{(-24)/(13) \cdot \sqrt{1 - ((-12)/(13))^(2)}}`

`= (1)/((-24)/(13)) \cdot \frac{1}{\sqrt{1 - (144)/(169)}}`

`= -(13)/(24) \cdot \frac{1}{\sqrt{(25)/(169)}}`

`= -(13)/(24) \cdot (1)/((5)/(13))`

`= -(13)/(24) \cdot (13)/(5)`

`= -(169)/(120)`