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14. \tan^{-1}(\sqrt3)-\sec^{-1}(-2)  is equal to

    (A)       \pi

    (B)   -\frac{\pi}{3}

    (C)       \frac{\pi}{3}

    (D)    \frac{2\pi}{3}

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;     

13. If \sin^{-1}x = y then

    (A)    0\leq y \leq \pi

    (B)    -\frac{\pi}{2} \leq y \leq \frac{\pi}{2}

    (C)     0 < y < \pi

    (D)     -\frac{\pi}{2} < y < \frac{\pi}{2}

Given if    then, As we know that the  can take values between  Therefore, . Hence answer choice (B) is correct.

Find the values of the following:

    12. \cos^{-1}\left(\frac{1}{2} \right ) + 2\sin^{-1}\left(\frac{1}{2} \right )

Here we have  let us assume that the value of ; then we have to find out the value of x +2y. Calculation of x :                 ,  Hence . Calculation of y :               .    Hence . The required sum will be =     .

Find the values of the following:

    11. \tan^{-1}(1) + \cos^{-1}\left(-\frac{1}{2} \right ) + \sin^{-1}\left(-\frac{1}{2} \right )

To find the values first we declare each term to some constant ;  , So we have ; or   Therefore,      So, we have  . Therefore , , So we have;    or   Therefore  Hence we can calculate the sum: .  

Find the principal values of the following:

    10. \textup{cosec}^{-1}(-\sqrt2)

Let us assume the value of  , then we have    or  . and the range of the principal values of   lies between  . hence the principal value of  is .  

Find the principal values of the following:

    9. \cos^{-1}\left(-\frac{1}{\sqrt2} \right )

Let us assume ; Then we have     or      , . And we know the range of principal values of  is . So, the only principal value which satisfies  is .

Find the principal values of the following:

    8. \cot^{-1}(\sqrt3)

Let us assume that  , then we can write in other way,    or    . Hence when     we have . and the range of principal values of  lies in . Then the principal value of  is   

Find the principal values of the following:

    7. \sec^{-1}\left (\frac{2}{\sqrt3}\right)

Let us assume that   then, we can also write it as; . Or   and the principal values lies between  . Hence we get only one principal value of   i.e., .  

Find the principal values of the following:

    6. \tan^{-1}(-1)

Given  so we can assume it to be equal to 'z';   ,        or                                                            And as we know the range of principal values of  from . As only one value z =  lies hence we have only one principal value that is .  

Find the principal values of the following:

    5. \cos^{-1}\left(-\frac{1}{2} \right )

Let us assume that   then, Easily we have;  or we can write it as:  as we know that the range of the principal values of   is . Hence  lies in the range it is a principal solution.

Find the principal values of the following:

    4. \tan^{-1}(-\sqrt3)

Let us assume that , then we have;  or   and as we know that the principal value of  is . Hence the only principal value of  when .

Find the principal values of the following:

    3. \textup{cosec}^{-1}(2)

Let us assume that , then we have; , or    . And we know the range of principal values is   Therefore the principal value of   is .

Find the principal values of the following:

    2. \cos^{-1}\left(\frac{\sqrt3}{2} \right )

So, let us assume that   then, Taking inverse both sides we get; ,  or             and as we know that the principal values of  is from [0,], Hence      when x =  .  Therefore, the principal value for is   .  

Find the principal values of the following:

    1.    \sin^{-1}\left(-\frac{1}{2} \right )

Let us assume that  ; Then taking inverse both sides we get; . As we know that the range of principal values of  is from . Hence only one possible value is  . Therefore, the principal value of  is .  

Find the principal values of the following : \sin^{-1}\left ( \frac{-1}{2} \right )

Let  We know, principle value range of  is   Principle value of  is 
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