How do you find the value of $sec [{cot}^{- 1} (- 6)]$ $sec [Cot^-1 (-6)]$ ?

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How do you find the value of $sec [{cot}^{- 1} (- 6)]$ $sec [Cot^-1 (-6)]$ ?

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Answer 1 · 2018-03-26T09:35:39

$- \frac{\sqrt{37}}{6}$ $-sqrt(37)/6$

Explanation:

We know that,
$(1) {cot}^{- 1} (- x) = π - {cot}^{- 1} x$ $color(red)((1)cot^-1(-x)=pi-cot^-1x$
$(2) sec (π - θ) = - sec θ$ $color(red)((2)sec(pi-theta)=-sectheta$
$(3) {cot}^{- 1} x = {tan}^{- 1} (\frac{1}{x}), x > 0$ $color(red)((3)cot^-1x=tan^-1(1/x), x > 0$
$(4) {tan}^{- 1} x = {cos}^{- 1} (\frac{1}{\sqrt{1 + x^{2}}})$ $color(red)((4)tan^-1x=cos^-1(1/(sqrt(1+x^2)))$
$(5) {cos}^{- 1} x = {sec}^{- 1} (\frac{1}{x})$ $color(red)((5)cos^-1x=sec^-1(1/x)$
$(6) sec ({sec}^{- 1} x) = x$ $color(red)((6)sec(sec^-1x)=x$
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 $color(red)((6)$

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How do you find the value of $sec [{cot}^{- 1} (- 6)]$ $sec [Cot^-1 (-6)]$ ?

1 Answer

Explanation:

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How do you find the value of $sec [{cot}^{- 1} (- 6)]$ $sec [Cot^-1 (-6)]$ ?