Question
Question: If \(\cot x-\tan x=2\), the generalized solution is (here, n is integer): (a) \(x=\dfrac{n\pi }{2}...
If cotx−tanx=2, the generalized solution is (here, n is integer):
(a) x=2nπ+8π
(b) x=4nπ+16π
(c) x=2nπ+2π
(d) x=nπ+4π
Solution
The equation given in the above problem is as follows: cotx−tanx=2, writing cotx=tanx1 in this trigonometric equation and we will get the quadratic in tanx and then rearrange the equation and get the value of tanx. After that find the angle at which you are getting that value of tanx. Let us suppose the angle that you are getting is θ which will look like tanx=tanθ then the general solution of this equation is equal to: x=nπ+θ.
Complete step by step solution:
The trigonometric equation given in the above problem is as follows:
cotx−tanx=2
Now, writing cotx=tanx1 in the above equation and we get,
tanx1−tanx=2
Taking tanx as L.C.M in the above equation we get,
tanx1−tan2x=2
On cross multiplying the above equation we get,
1−tan2x=2tanx
Now, dividing (1−tan2x) on both the sides of the above equation we get,
1−tan2x1−tan2x=1−tan2x2tanx
In the L.H.S of the above equation, (1−tan2x) will be cancelled out from the numerator and the denominator and we get,
1=1−tan2x2tanx …………. (1)
We know that there is a double angle identity of tangent which is equal to:
tan2x=1−tan2x2tanx
Using the above relation in eq. (1) we get,
1=tan2x
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 tan4π then the above equation will look like:
tan4π=tan2x
Rearranging the above equation we get,
tan2x=tan4π
We know that the general solution for tanx=tanθ is equal to:
x=nπ+θ
Now, using the above general solution we can write the general solution of the above equation.
2x=nπ+4π
Dividing 2 on both the sides of the above equation we get,
x=2nπ+8π
From the above, we got the general solution for the above equation as: x=2nπ+8π.
So, the correct answer is “Option A”.
Note: To solve the above problem, you should know how to write the general solution for tanx 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.