Question

How do you find the value of `sec [Cot^-1 (-6)]`?

Answer 1

`-sqrt(37)/6`

Explanation:

We know that,
`color(red)((1)cot^-1(-x)=pi-cot^-1x`
`color(red)((2)sec(pi-theta)=-sectheta`
`color(red)((3)cot^-1x=tan^-1(1/x), x > 0`
`color(red)((4)tan^-1x=cos^-1(1/(sqrt(1+x^2)))`
`color(red)((5)cos^-1x=sec^-1(1/x)`
`color(red)((6)sec(sec^-1x)=x`
Here,

`sec(cot^-1(-6))=sec(pi-cot^-1(6))...to`Apply `color(red)((1)`

`=-sec(cot^-1(6)).......to`Apply `color(red)((2)`

`=-sec(tan^-1(1/6)).........to`Apply `color(red)((3)`

`=-sec(cos^-1(1/(sqrt(1+(1/6)^2))))......to`Apply `color(red)((4)`

`=-sec(cos^-1(1/(sqrt(1+1/36))))`

`=-sec(cos^-1(6/(sqrt(37))))`

`=-sec(sec^-1(sqrt(37)/6)).......to`Apply `color(red)((5)`

`=-sqrt(37)/6...........to`Apply `color(red)((6)`