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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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