Q&A - Ask Doubts and Get Answers

Sort by :
Clear All
Q

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.

Prove the following:

    2. 3\cos^{-1} x = \cos^{-1}(4x^3 - 3x), \;\;x\in\left[\frac{1}{2},1 \right ]

Given to prove . Take   or ; Then we have; R.H.S. =        =  =  =  = L.H.S Hence Proved.

Prove the following:

    1. 3\sin^{-1}x = \sin^{-1}(3x - 4x^3),\;\;x\in\left[-\frac{1}{2},\frac{1}{2} \right ]

Given to prove:       where,     . Take   or   Take R.H.S value   =  =                                         =  =    =   L.H.S

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.
Exams
Articles
Questions