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Given the equation:  we can migrate the  term to the R.H.S. then we have; or                                ............................(1) from     Take       or    . So, we conclude that; Therefore we can put the value of  in equation (1)  we get, Putting x= sin y, in the above equation; we have then,   So, we have the solution;     Therefore we have . When we have , we can see that...
Given equation is : L.H.S can be written as;   Using the formula So, we have         Hence the value of .
Given equation ; Using the formula: We can write So, we can equate; that implies that . or          or    Hence we have solution .  
We have to solve the given equation:   Take as common in L.H.S, or     from      Now, assume,   Then, Therefore we have now, So we have L.H.S then That is equal to R.H.S. Hence proved.          
By using the Hint we will put ; we get then,    dividing numerator and denominator by , we get,       using the formula     As L.H.S = R.H.S Hence proved  
Given that By observing we can rationalize the fraction         We get then, Therefore we can write it as;   As L.H.S. = R.H.S. Hence proved.    
By observing the square root we will first put . Then, we have or, R.H.S. . L.H.S. hence L.H.S. = R.H.S proved.  
Applying the formlua:     on two parts. we will have,     Hence it s equal to R.H.S Proved.
Taking R.H.S; We have Converting sin and cos terms in tan forms: Let    and   now, we have    or                        ............(1) Now,                              ................(2) Now, Using (1) and (2) we get, R.H.S.     as we know so, equal to L.H.S Hence proved.    
Converting all terms in tan form; Let ,    and . now, converting all the terms:   or   We can write it in tan form as: . or            ................(1)      or   We can write it in tan form as: or          ......................(2) Similarly, for ; we have      .............(3) Using (1) and (2) we have L.H.S On applying We have,                                  ...........[Using...
Take   and     and then we have, Then we can write it as:     or                                       ...............(1) Now,    So,                          ...................(2) Also we have similarly;   Then,              ...........................(3) Now, we have L.H.S     so, using (1) and (2) we get,                      or we can write it as;    =  R.H.S. Hence proved.  
Taking    then, Therefore we have-               .............(1). , Then,                     .............(2). So, we have now, L.H.S. using equations (1) and (2) we get,                            =  R.H.S.
To prove: ; Assume that    then we have . or  Therefore we have   Now, We can write L.H.S as        as we know           L.H.S = R.H.S  
We have given ; so, as we know  So, here we have . Therefore we can write  as:               .  
If   then   , which is principal value of . So, we have       Therefore we have, .
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).
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.  
As we know that  if  and is principal value range of . In this case , hence we have then,     Hence the correct answer is  (B).
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