In the previous section, we saw the first property of inverse trigonometric functions. In this section, we will see the second and third properties.
Property II
• This has 3 parts:
$\begin{array}{cc}{} &{\text{(i)}} &{\sin^{-1}\left(-x \right)} {}={} &{-\sin^{-1}x}, &{x \in [-1,1]}\\
{} &{\text{(ii)}} &{\tan^{-1}\left(-x \right)} {}={} &{-\tan^{-1}x}, &{x \in R}\\
{} &{\text{(iii)}} &{\csc^{-1}\left(-x \right)} {}={} &{-\csc^{-1}x}, &{|x| \ge 1}\\
\end{array}$
• Let us prove part (i):
$\begin{array}{ll}{} &{\text{Let}~\sin^{-1} \left(-x \right)} & {~=~} &{y~\color{magenta}{\text{- - - (A)}}} &{} \\
{\implies} &{\sin y} & {~=~} &{-x} &{} \\
{\implies} &{- \sin y} & {~=~} &{x} &{} \\
{\implies} &{\sin (-y)} & {~=~} &{x~\color{magenta}{\text{- - - (B)}}} &{} \\
{\implies} &{\sin^{-1} x} & {~=~} &{-y} &{} \\
{\implies} &{-\sin^{-1} x} & {~=~} &{y} &{} \\
{\implies} &{-\sin^{-1} x} & {~=~} &{\sin^{-1} \left(-x \right)~\color{magenta}{\text{- - - (C)}}} &{} \\
\end{array}$
◼ Remarks:
• In line (A), we assume that $\sin^{-1}(-x)$ = y
• In line B, we use the identity 1: sin (-𝜃) = - sin 𝜃
♦ List of identities can be seen here.
• In line C, we substitute for y, using the assumption in (A).
• Let us prove part (ii):
$\begin{array}{ll}{} &{\text{Let}~\tan^{-1} \left(-x \right)} & {~=~} &{y~\color{magenta}{\text{- - - (A)}}} &{} \\
{\implies} &{\tan y} & {~=~} &{-x} &{} \\
{\implies} &{- \tan y} & {~=~} &{x} &{} \\
{\implies} &{\tan (-y)} & {~=~} &{x~\color{magenta}{\text{- - - (B)}}} &{} \\
{\implies} &{\tan^{-1} x} & {~=~} &{-y} &{} \\
{\implies} &{-\tan^{-1} x} & {~=~} &{y} &{} \\
{\implies} &{-\tan^{-1} x} & {~=~} &{\tan^{-1} \left(-x \right)~\color{magenta}{\text{- - - (C)}}} &{} \\
\end{array}$
◼ Remarks:
• In line (A), we assume that $\tan^{-1}(-x)$ = y
• In line B, we use the identities 1 and 2 to get: tan (-𝜃) = - tan 𝜃
♦ List of identities can be seen here.
• In line C, we substitute for y, using the assumption in (A).
• Let us prove part (iii):
$\begin{array}{ll}{} &{\text{Let}~\csc^{-1} \left(-x \right)} & {~=~} &{y~\color{magenta}{\text{- - - (A)}}} &{} \\
{\implies} &{\csc y} & {~=~} &{-x} &{} \\
{\implies} &{- \csc y} & {~=~} &{x} &{} \\
{\implies} &{\frac{-1}{\sin y}} & {~=~} &{x} &{} \\
{\implies} &{-\sin y} & {~=~} &{\frac{1}{x}} &{} \\
{\implies} &{\sin (-y)} & {~=~} &{\frac{1}{x}~\color{magenta}{\text{- - - (B)}}} &{} \\
{\implies} &{\sin^{-1} \left(\frac{1}{x} \right)} & {~=~} &{-y} &{} \\
{\implies} &{\csc^{-1} x } & {~=~} &{-y~\color{magenta}{\text{- - - (C)}}} &{} \\
{\implies} &{\csc^{-1} x} & {~=~} &{-\csc^{-1} \left(-x \right)~\color{magenta}{\text{- - - (D)}}} &{} \\
{\implies} &{-\csc^{-1} x} & {~=~} &{\csc^{-1}
\left(-x \right)} &{} \\
\end{array}$
◼ Remarks:
• In line (A), we assume that $\csc^{-1}(-x)$ = y
• In line B, we use the identity 1: sin (-𝜃) = - sin 𝜃
♦ List of identities can be seen here.
• In line C, we use part (i) of property I
• In line D, we substitute for y, using the assumption in (A).
Property III
• This has 3 parts:
$\begin{array}{cc}{} &{\text{(i)}} &{\cos^{-1}\left(-x \right)} {}={} &{\pi-\cos^{-1}x}, &{x \in [-1,1]}\\
{} &{\text{(ii)}} &{\sec^{-1}\left(-x \right)} {}={} &{\pi-\sec^{-1}x}, &{|x| \ge 1}\\
{} &{\text{(iii)}} &{\cot^{-1}\left(-x \right)} {}={} &{\pi-\cot^{-1}x}, &{x \in R}\\
\end{array}$
• Let us prove part (i):
◼ Remarks:
• In line (A), we assume that $\cos^{-1}(-x)$ = y
• In line B, we use the identity 1: cos (π-𝜃) = - cos 𝜃
♦ List of identities can be seen here.
• In line C, we substitute for y, using the assumption in (A).
• Let us prove part (ii):
◼ Remarks:
• In line (A), we assume that $\sec^{-1}(-x)$ = y
• In line B, we use the identity 1: cos (π-𝜃) = - cos 𝜃
♦ List of identities can be seen here.
• In line C, we use part (ii) of property I
• In line D, we substitute for y, using the assumption in (A).
• Let us prove part (iii):
◼ Remarks:
• In line (A), we assume that $\cot^{-1}(-x)$ = y
• In line B, we use the identities 9(c) and 9(d) to get: cot (π-𝜃) = - cot 𝜃
♦ List of identities can be seen here.
• In line C, we substitute for y, using the assumption in (A).
In the next section, we
will see fourth and fifth properties.
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