In the previous section, we saw the second and third properties of inverse trigonometric functions. In this section, we will see the fourth and fifth properties.
Property IV
• This has 3 parts:
$\begin{array}{cc}{} &{\text{(i)}} &{\sin^{-1}x + \cos^{-1}x} {}={} &{\frac{\pi}{2}}, &{x \in [-1,1]}\\
{} &{\text{(ii)}} &{\tan^{-1}x + \cot^{-1}x} {}={} &{\frac{\pi}{2}}, &{x \in R}\\
{} &{\text{(iii)}} &{\csc^{-1}x + \sec^{-1}x} {}={} &{\frac{\pi}{2}}, &{|x| \ge 1}\\
\end{array}$
• Let us prove part (i):
◼ Remarks:
• In line A, we assume that $\sin^{-1}(x)$ = y
• In line B, we use the identity 5: $\sin \theta = \cos \left(\frac{\pi}{2} - \theta \right)$
♦ List of identities can be seen here.
• In line C, we take the inverse.
• In line D, we use the information from A and C.
• Let us prove part (ii):
◼ Remarks:
• In line A, we assume that $\tan^{-1}(x)$ = y
• In line B, we use the identity 5: $\tan \theta = \cot \left(\frac{\pi}{2} - \theta \right)$
♦ List of identities can be seen here.
• In line C, we take the inverse.
• In line D, we use the information from A and C.
• Let us prove part (iii):
◼ Remarks:
• In line A, we assume that $\tan^{-1}(x)$ = y
• In line B, we use the identity 5: $\tan \theta = \cot \left(\frac{\pi}{2} - \theta \right)$
♦ List of identities can be seen here.
• In line C, we take the inverse.
• In line D, we use the information from A and C.
Property V
• This has 3 parts:
$\begin{array}{cc}{} &{\text{(i)}} &{\tan^{-1}x + \tan^{-1}y} &{}={} &{\tan^{-1}\frac{x+y}{1-xy}}, &{xy < 1}\\
{} &{\text{(ii)}} &{\tan^{-1}x - \tan^{-1}y} &{}={} &{\tan^{-1}\frac{x-y}{1+xy}}, &{xy > -1}\\
{} &{\text{(iii)}} &{2\tan^{-1}x} &{}={} &{\tan^{-1}\frac{2x}{1-x^2}}, &{-1 < x < 1}\\
\end{array}$
• Let us prove part (i):
◼ Remarks:
• In line A, we assume that $\tan^{-1}(x) = \theta$
• In line B, we assume that $\tan^{-1}(y) = \phi$
• In line C, we use the identity 10.
♦ List of identities can be seen here.
• In line D, we make the substitutions based on A and B.
• In line E, we make the reverse substitutions based on A and B.
• Let us prove part (ii):
◼ Remarks:
• In line A, we assume that $\tan^{-1}(x) = \theta$
• In line B, we assume that $\tan^{-1}(y) = \phi$
• In line C, we use the identity 11.
♦ List of identities can be seen here.
• In line D, we make the substitutions based on A and B.
• In line E, we make the reverse substitutions based on A and B.
Alternate method for proving part (ii):
◼ Remarks:
• In line A, we use part (i).
• In line B, we put (-y) in place of y.
• In line L.H.S of line C, we use part (ii) of property II.
• Let us prove part (iii):
◼ Remarks:
• In line A, we use part (i).
• In line B, we put (-y) in place of y.
• In line L.H.S of line C, we use part (ii) of property II.
In the next section, we
will see the sixth property.
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