Question

How do you solve $ {\tan ^2}x = \tan x $ ?

Answer 1

Hint: Every trigonometric function and formulae are designed on the basis of three primary ratios. Sine, Cosine and tangents are these ratios in trigonometry based on Perpendicular, Hypotenuse and Base of a right triangle . In order to calculate the angles sin , cos and tan functions . According to the formula of tan = $ \dfrac{{\sin \theta }}{{\cos \theta }} $ , we will find the value of x . Also we will find x by transposing as required by the question.

Complete step-by-step answer:
We are given the question as $ {\tan ^2}x = \tan x $ , we will subtract the function tan x from both the sides L. H. S. and R. H. S.
 $ {\tan ^2}x - \tan x = \tan x - \tan x $
In R. H. S. there it is left with zero , so that we can solve –
 $ {\tan ^2}x - \tan x = 0 $
 $ \tan x(\tan x - 1) = 0 $
Here we are taking the common from the L. H. S. to make it simpler and value of x can be determined ,
Like the quadratic equations we can solve the x as ,
According to the formula of tan = $ \dfrac{{\sin \theta }}{{\cos \theta }} $ , we will find the value of x as tan x can be equated with zero and the tan x - 1 can be equated with zero separately .
 $
  \tan x = 0 and \\
  \tan x - 1 = 0 \\
  \tan x = 1 \;
  $
According to the formula of tan = $ \dfrac{{\sin \theta }}{{\cos \theta }} $ ,
 $ \dfrac{{\sin x}}{{\cos x}} = 0 $ or $ \dfrac{{\sin \theta }}{{\cos \theta }} = 1 $
Now , we will multiply by cos x on both the sides L. H. S. and R. H. S. , we get -
 $ \sin x = 0 $ or $ \sin x = \cos x $
Now we have to calculate for angle x for which we will give a general solution ,
 $ x = k\pi $ or $ x = \dfrac{\pi }{4} + k\pi $ where $ k \in \mathbb{Z} $
This is the final answer .
So, the correct answer is “ $ x = k\pi $ or $ x = \dfrac{\pi }{4} + k\pi $”.

Note: Even Function – A function $ f(x) $ is said to be an even function ,if $ f( - x) = f(x) $ for all x in its domain.
Odd Function – A function $ f(x) $ is said to be an even function ,if $ f( - x) = - f(x) $ for all x in its domain.
Periodic Function= A function $ f(x) $ is said to be a periodic function if there exists a real number T > 0 such that $ f(x + T) = f(x) $ for all x.
If T is the smallest positive real number such that $ f(x + T) = f(x) $ for all x, then T is called the fundamental period of $ f(x) $ .
Since $ \sin \,(2n\pi + \theta ) = \sin \theta $ for all values of $ \theta $ and n $ \in $ N.
It should be very clear that \[sin\left( {A + B} \right)\]is not equal to $ \sin A + \sin B $ .