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Thursday, January 25, 2024

18.11 - Solved Examples

In the previous section, we saw the sixth property of inverse trigonometric functions. We saw two solved examples also. In this section, we will see a few more solved examples.

Solved example 18.5
Express tan1(cosx1sinx) in the simplest form
Solution:


◼ Remarks:
• Line 1:
   ♦ For the numerator, we use identity 14.
   ♦ For the denominator, we use the identity 15.
   ♦ List of identities can be seen here.
• Line 2:
   ♦ For the numerator, we use identity (a2 - b2) = (a+b)(a-b).
   ♦ For the denominator, we use the identity a2-2ab+b2 = (a-b)2.
• Line 3: We divide both numerator and denominator by cosx2.
• Line 5: We put tanπ4 in the place of 1.
• Line 6: We use part (i) of property V.
• Line 7: We apply the fact that, function of the inverse function will give the same output as input.

Alternate method:


◼ Remarks:
• Line 1:
   ♦ For the numerator, we use identity 6.
   ♦ For the denominator, we use the identity 5.
   ♦ List of identities can be seen here.
• Line 3:
   ♦ For the numerator, we use identity 15.
   ♦ For the denominator, we use the identity 14.
• Line 5: We use the identities 5 and 6.
• Line 6: We apply the fact that, function of the inverse function will give the same output as input.

Solved example 18.6
Write cot1(1x21),|x|>1 in the simplest form.
Solution:


◼ Remarks:
• Line 1: We assume that, the given expression is equal to y. So our aim is to find y.
• Line 4: We use a basic identity.
• Line 7: The square root of x2 is ±x. But in this problem, it is given that, |x| > 1. So we need not consider the -ve sign.

Solved example 18.7
Prove that  tan1x+tan12x1x2=tan1(3xx313x2),|x|<13.
Solution:

◼ Remarks:
• Line 1: We apply part (i) of property V.

Solved example 18.7
Find the value of cos(sec1+csc1x), |x|1.
Solution:

◼ Remarks:
• Line 1: We use parts (i) and (ii) of property I.
• Line 2: We use part (i) of property IV.


The link below, gives some more solved examples:

Exercise 18.2 


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