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Find the value of the following:

2.

We have given ; so, as we know  So, here we have . Therefore we can write  as:               .

21.   is equal to

(A)

(B)

(C)    0

(D)

We have ; finding the value of  : Assume  then,   and the range of the principal value of  is . Hence, principal value is  Therefore  and  so, we have now, or,  Hence the answer is option  (B).

20.  is equal to

(A)

(B)

(C)

(D)

Solving the inner bracket of ;   or Take   then,   and we know the range of principal value of  Therefore we have . Hence,  Hence the correct answer is D.

19.  is equal to

(A)

(B)

(C)

(D)

As we know that  if  and is principal value range of . In this case , hence we have then,     Hence the correct answer is  (B).

Find the values of each of the expressions in Exercises 16 to 18.

18.

Given that   we can take , then   or                    We have similarly; Therefore we can write              from

Find the values of each of the expressions in Exercises 16 to 18.

17.

As we know  If  which is the principal value range of . So, as in ; Hence we can write    as :  =  Where  and

Find the values of each of the expressions in Exercises 16 to 18.

16.

Given ; We know that   If the value of x belongs to   then we get the principal values of . Here,  We can write   is as: =   =   where

15. If , then find the value of .

Using the identity , We can find the value of x; So,  on applying, =  =     or   , Hence, the possible values of x are .

14. If , then find the value of .

As we know the identity;  . it will just hit you by practice to apply this. So,      or    , we can then write , putting in above equation we get;                            =    Ans.

Find the values of each of the following:

13.  and

Taking the value   or    and   or  then we have, = , =  Then,       Ans.

Find the values of each of the following:

12.

We have to find the value of   As we know    so, Equation reduces to .

Find the values of each of the following:

11.

Given equation: So, solving the inner bracket first, we take the value of  Then we have, Therefore, we can write .   .

Write the following functions in the simplest form:

10.

Given  Here we can take  So,     will become; and as ; hence the simplest form is .

Write the following functions in the simplest form:

9.

Given that  Take    or      and putting it in the equation above;      is the simplest form.

Write the following functions in the simplest form:

8.

Given    where So,    Taking  common from numerator and denominator. We get:  =          as,  =  is the simplest form.

Write the following functions in the simplest form:

7.

Given that  We have in inside the root the term :  Put    and    , Then we have, Hence the simplest form is

Write the following functions in the simplest form:

6.

Given that  Take   or          =   =

Write the following functions in the simplest form:

5.

We have  Take        is the simplified form.

Prove the following:

4.

Given to prove  Then taking L.H.S. We have                                                              =  R.H.S. Hence proved.

.

Prove the following:

3.

Given to prove  We have L.H.S              =   R.H.S Hence proved.
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