Question

If $\tan (A - B) = 1,$ $\sec (A + B) = \dfrac{2}{{\sqrt 3 }}$, then the smallest positive value of $B$ is?
A.$\dfrac{{25\pi }}{{24}}$
B.$\dfrac{{19\pi }}{{24}}$
C. $\dfrac{{13\pi }}{{24}}$
D. $\dfrac{{11\pi }}{{24}}$

Answer 1

Hint : In this type we find the value of $A + B$ and $A - B$ and solving this system of equations we can find the value of $B$. In this particular question we will have cases, because they asked only the positive value. If we get a positive value it will be no problem. If we get negative value we move to further case which we have done below:

Complete step-by-step answer :
Case (1):
Given, $\tan (A - B) = 1,$we can be rewrite it as,
$\tan (A - B) = \tan \left( {\dfrac{\pi }{4}} \right),$
Cancelling on both side we get,
$\Rightarrow A - B = \dfrac{\pi }{4}$
Similarly $\sec (A + B) = \dfrac{2}{{\sqrt 3 }}$ we can be rewrite it as,
$\sec (A + B) = \sec \left( {\dfrac{\pi }{6}} \right)$
Cancelling on both side,
$\Rightarrow A + B = \dfrac{\pi }{6}$
We subtracted because we need the value of $B$ only, if added above we will get the value of $A$.
So Subtracting, $A - B = \dfrac{\pi }{4}$ with $A + B = \dfrac{\pi }{6}$we get
$\Rightarrow A - B - A - B = \dfrac{\pi }{4} - \dfrac{\pi }{6}$
$A$ Will get cancels out,
Taking L.C.M. and simplifying we get,
$\Rightarrow - 2B = \dfrac{\pi }{{12}}$
Divided by $- 2$ on both sides,
$\Rightarrow B = - \dfrac{\pi }{{24}}$
But $B$ cannot be negative, we need only positive value, we change any one of the angles and continue the same procedure.
Case (2):
Changing the angle,
So we take $\sec \left( {\dfrac{\pi }{6}} \right)$ as $\sec \left( {2\pi - \dfrac{\pi }{6}} \right)$.
Because we know $\sec \left( {2\pi - \theta } \right) = \sec \theta$.
Then, $\sec (A + B) = \sec \left( {2\pi - \dfrac{\pi }{6}} \right)$
Cancelling on both sides,
$\Rightarrow A + B = 2\pi - \dfrac{\pi }{6}$
$\Rightarrow A + B = \dfrac{{11\pi }}{6}$.
Now we have, $A - B = \dfrac{\pi }{4}$ and $A + B = \dfrac{{11\pi }}{6}$, subtracting we get,
$\Rightarrow A - B - A - B = \dfrac{\pi }{4} - \dfrac{{11\pi }}{6}$
$\Rightarrow - 2B = - \dfrac{{19\pi }}{{12}}$
$\Rightarrow B = \dfrac{{19\pi }}{{24}}$.
So, the correct answer is “Option B”.

Note : Since they asked us to find a positive value of $B$ at first we get negative value while solving the equations in case (I) so we proceed to case (2) where we get a positive value. If they asked only the value of $B$ and not the positive value we can stop the solution at case (I) only. . If we don’t get positive value even in case (2). We move to case (3) by changing the angle again.