18.9 - Properties IV and V

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.

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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.

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In the next section, we will see the sixth property.

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