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Given that   we can take , then   or                    We have similarly; Therefore we can write              from
As we know  If  which is the principal value range of . So, as in ; Hence we can write    as :  =  Where  and
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
Using the identity , We can find the value of x; So,  on applying, =  =     or   , Hence, the possible values of x are .
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.
Taking the value   or    and   or  then we have, = , =  Then,       Ans.
We have to find the value of   As we know    so, Equation reduces to .
Given equation: So, solving the inner bracket first, we take the value of  Then we have, Therefore, we can write .   .
Given  Here we can take  So,     will become; and as ; hence the simplest form is .
Given that  Take    or      and putting it in the equation above;      is the simplest form.
Given    where So,    Taking  common from numerator and denominator. We get:  =          as,  =  is the simplest form.
Given that  We have in inside the root the term :  Put    and    , Then we have, Hence the simplest form is
Given that  Take   or          =   =
We have  Take        is the simplified form.
Given to prove  Then taking L.H.S. We have                                                              =  R.H.S. Hence proved.
Given to prove  We have L.H.S              =   R.H.S Hence proved.
Given to prove . Take   or ; Then we have; R.H.S. =        =  =  =  = L.H.S Hence Proved.
Given to prove:       where,     . Take   or   Take R.H.S value   =  =                                         =  =    =   L.H.S
Let us assume the values of  be 'x'  and   be 'y'. Then we have;      or       or     or  .  and      or                 or        also the ranges of the principal values of  and     are . and     respectively.  we have then;
Given if    then, As we know that the  can take values between  Therefore, . Hence answer choice (B) is correct.
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