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Find the value of the following:

2. $\tan^{-1}\left(\tan\frac{7\pi}{6} \right )$

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

21. $\tan^{-1}\sqrt3 - \cot^{-1}(-\sqrt3)$  is equal to

(A)    $\pi$

(B)    $-\frac{\pi}{2}$

(C)    0

(D)    $2\sqrt3$

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. $\sin\left(\frac{\pi}{3} -\sin^{-1}\left(-\frac{1}{2} \right ) \right )$ is equal to

(A)    $\frac{1}{2}$

(B)    $\frac{1}{3}$

(C)    $\frac{1}{4}$

(D)    $1$

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. $\cos^{-1}\left(\cos\frac{7\pi}{6} \right )$ is equal to

(A)    $\frac{7\pi}{6}$

(B)    $\frac{5\pi}{6}$

(C)    $\frac{\pi}{3}$

(D)    $\frac{\pi}{6}$

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. $\tan\left(\sin^{-1}\frac{3}{5}+\cot^{-1}\frac{3}{2} \right )$

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. $\tan^{-1}\left (\tan\frac{3\pi}{4} \right )$

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. $\sin^{-1}\left (\sin\frac{2\pi}{3} \right )$

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 $\tan^{-1}\frac{x-1}{x-2} + \tan^{-1}\frac{x+1}{x+2} =\frac{\pi}{4}$, then find the value of $x$.

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

14. If $\sin\left(\sin^{-1}\frac{1}{5} + \cos ^{-1}x \right ) =1$, then find the value of $x$.

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. $\tan \frac{1}{2}\left[\sin^{-1}\frac{2x}{1+x^2} + cos^{-1}\frac{1-y^2}{1+y^2} \right ],\;\;|x|<1,\;y>0$ and $xy<1$

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

Find the values of each of the following:

12. $\cot(\tan^{-1}a + \cot^{-1}a)$

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

Find the values of each of the following:

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

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. $\tan^{-1}\left(\frac{3a^2x -x^3}{a^3 - 3ax^2} \right ),\;\;a>0\;\;;\;\;\frac{-a}{\sqrt3} < x < \frac{a}{\sqrt3}$

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

Write the following functions in the simplest form:

9. $\tan^{-1} \frac{x}{\sqrt{a^2 - x^2}}, \;\; |x| < a$

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

Write the following functions in the simplest form:

8. $\tan^{-1}\left(\frac{\cos x -\sin x }{\cos x + \sin x} \right ),\;\; \frac{-\pi}{4} < x < \frac{3\pi}{4}$

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. $\tan^{-1}\left(\sqrt{\frac{1-\cos x}{1 + \cos x}} \right ),\;\; 0< x < \pi$

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. $\tan^{-1} \frac{1}{\sqrt{x^2 -1}},\;\; |x| > 1$

Given that  Take   or          =   =

Write the following functions in the simplest form:

5. $\tan^{-1}\frac{\sqrt{1 + x^2}- 1}{x},\;\;x\neq 0$

We have  Take        is the simplified form.

Prove the following:

4. $2\tan^{-1} \frac{1}{2} + \tan^{-1}\frac{1}{7} = \tan^{-1}\frac{31}{17}$

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

.

Prove the following:

3. $\tan^{-1}\frac{2}{11} + \tan^{-1}\frac{7}{24} = \tan^{-1}\frac{1}{2}$

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