Question

If $\cot x-\tan x=2$, the generalized solution is (here, n is integer):
(a) $x=\dfrac{n\pi }{2}+\dfrac{\pi }{8}$
(b) $x=\dfrac{n\pi }{4}+\dfrac{\pi }{16}$
(c) $x=2n\pi +\dfrac{\pi }{2}$
(d) $x=n\pi +\dfrac{\pi }{4}$

Answer 1

Hint: The equation given in the above problem is as follows: $\cot x-\tan x=2$, writing $\cot x=\dfrac{1}{\tan x}$ in this trigonometric equation and we will get the quadratic in $\tan x$ and then rearrange the equation and get the value of $\tan x$. After that find the angle at which you are getting that value of $\tan x$. Let us suppose the angle that you are getting is $\theta $ which will look like $\tan x=\tan \theta $ then the general solution of this equation is equal to: $x=n\pi +\theta $.

Complete step by step solution:
The trigonometric equation given in the above problem is as follows:
$\cot x-\tan x=2$
Now, writing $\cot x=\dfrac{1}{\tan x}$ in the above equation and we get,
$\dfrac{1}{\tan x}-\tan x=2$
Taking $\tan x$ as L.C.M in the above equation we get,
$\dfrac{1-{{\tan }^{2}}x}{\tan x}=2$
On cross multiplying the above equation we get,
$1-{{\tan }^{2}}x=2\tan x$
Now, dividing $\left( 1-{{\tan }^{2}}x \right)$ on both the sides of the above equation we get,
$\dfrac{1-{{\tan }^{2}}x}{1-{{\tan }^{2}}x}=\dfrac{2\tan x}{1-{{\tan }^{2}}x}$
In the L.H.S of the above equation, $\left( 1-{{\tan }^{2}}x \right)$ will be cancelled out from the numerator and the denominator and we get,
$1=\dfrac{2\tan x}{1-{{\tan }^{2}}x}$ …………. (1)
We know that there is a double angle identity of tangent which is equal to:
$\tan 2x=\dfrac{2\tan x}{1-{{\tan }^{2}}x}$
Using the above relation in eq. (1) we get,
$1=\tan 2x$
Now, to find the general solution, in the above equation, both sides contain tangent term so in the L.H.S of the above equation, we can write 1 as $\tan \dfrac{\pi }{4}$ then the above equation will look like:
$\tan \dfrac{\pi }{4}=\tan 2x$
Rearranging the above equation we get,
$\tan 2x=\tan \dfrac{\pi }{4}$
We know that the general solution for $\tan x=\tan \theta $ is equal to:
$x=n\pi +\theta $
Now, using the above general solution we can write the general solution of the above equation.
$2x=n\pi +\dfrac{\pi }{4}$
Dividing 2 on both the sides of the above equation we get,
$x=\dfrac{n\pi }{2}+\dfrac{\pi }{8}$
From the above, we got the general solution for the above equation as: $x=\dfrac{n\pi }{2}+\dfrac{\pi }{8}$.

So, the correct answer is “Option A”.

Note: To solve the above problem, you should know how to write the general solution for $\tan x$ otherwise you could not solve this problem. Also, you should know the double angle identity of tangent. If you miss any of the two concepts then it will be very hard for you to solve this problem so make sure you have properly understood these concepts.