Question

write
${cosec}^{ - 1} ( - x) \: and \: {sec}^{ - 1} ( - x)$
in terms of
${cosec}^{ - 1} (x) \: and \: {sec}^{ - 1} (x)$

Answer 1

Answer:

Step-by-step explanation:

Hello there,

Inverse cosecant(-x) = (-1)(cosecant(x))

Inverse secant(-x) = π - Inverse secant (x)

They are odd functions.

i.e. f(-x) = -f(x)

Hope you find my answer useful.

Harith

Answer 2

it's just based on basic inverse trigonometry.

$sec^{-1}(-x)=\pi-sec^{-1}x\\\\cosec^{-1}(-x)=-cosec^{-1}x$

Let's see how's this possible.

Let $sec^{-1}(-x)=y$.....(1)

then, $-x=secy$

or, $x=-secy=sec(\pi-y)$

or, $sec^{-1}=\pi-y$

from equation (1),

$sec^{-1}x=\pi-sec^{-1}(-x)$

hence, $sec^{-1}(-x)=\pi-sec^{-1}x$

now, for $cosec^{-1}(-x)=-cosec^{-1}x$

Let us consider, $cosec^{-1}(-x)=z$.....(1)

or, $-x=cosecz\implies x=-cosecz$

we know, cosec(-A) = -cosecA

so, you can use , -cosecz = cosec(-z)

now, x = cosec(-z)

$cosec^{-1}x=-z$

from equation (1),

$cosec^{-1}x=-cosec^{-1}(-x)$

and hence, $cosec^{-1}(-x)=-cosec^{-1}x$