Sunday, January 21, 2024

18.10 - Property VI

In the previous section, we saw the fourth and fifth properties of inverse trigonometric functions. In this section, we will see the sixth property.

Property VI
• This has 3 parts:

(i)2tan1x=sin12x1+x2,|x|1(ii)2tan1x=cos11x21+x2,x0(iii)2tan1x=tan12x1x2,1<x<1

• Let us prove part (i):


◼ Remarks:
• In line A, we assume that tan1(x) = y
• In line B, we use the identity 15.
   ♦ List of identities can be seen here.
• In line C, we substitute for y from A.

• Let us prove part (ii):


◼ Remarks:
• In line A, we assume that tan1(x) = y
• In line B, we use the identity 14.
   ♦ List of identities can be seen here.
• In line C, we substitute for y from A.

• Let us prove part (iii). We already saw the proof in property V. Here we will see an alternate method:


◼ Remarks:
• In line A, we assume that tan1(x) = y
• In line B, we use the identity 16.
   ♦ List of identities can be seen here.
• In line C, we substitute for y from A.


Now we will see some solved examples.

Solved example 18.3
Show that
(i)sin1(2x1x2)=2sin1x,12x12(ii)sin1(2x1x2)=2cos1x,12x1
Solution:

Part (i):


◼ Remarks:
• Line 1: We assume that sin1(x) = y
• Line 3: We use the identity sin2𝜃 + cos2𝜃 = 1.
• Line 6: We use the identity 15.
   ♦ List of identities can be seen here.
• Line 8: We take the inverse.
• Line 9: We substitute for y based on the assumption in (1).

Part (ii):

 

◼ Remarks:
• Line 1: We assume that cos1(x) = y
• Line 3: We use the identity sin2𝜃 + cos2𝜃 = 1.
• Line 6: We use the identity 15.
   ♦ List of identities can be seen here.
• Line 8: We take the inverse.
• Line 9: We substitute for y based on the assumption in (1).

Solved example 18.4
Show that tan112+tan1211=tan134
Solution:


◼ Remarks:
• Line 1: We use part 2 of property 5.
• Line 2: We substitute the known values.


In the next section, we will see a few more solved examples.

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