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 = ${\displaystyle \frac{\sin \theta }{\cos \theta }}$ , 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 = ${\displaystyle \frac{\sin \theta }{\cos \theta }}$ , 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 .
 $$\begin{gathered} \tan x=0and \\ \tan x-1=0 \\ \tan x=1 \end{gathered}$$
According to the formula of tan = ${\displaystyle \frac{\sin \theta }{\cos \theta }}$ ,
 ${\displaystyle \frac{\sin x}{\cos x}}=0$ or ${\displaystyle \frac{\sin \theta }{\cos \theta }}=1$
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 $x={\displaystyle \frac{\pi }{4}}+k\pi$ where $k\in \mathbb{Z}$
This is the final answer .
So, the correct answer is “ $x=k\pi$ or $x={\displaystyle \frac{\pi }{4}}+k\pi$”.

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 $\sin A+\sin B$ .