Question

How do you solve $\tan x\sec x=0$ ?

Answer 1

In this question, we have to find the value of x of a trigonometric equation. Since the equation consists of trigonometric functions, thus we will use the trigonometric formulas to get the required result. We will first change the tan and sec function into sin and cos function, that is we will apply the trigonometric formula $\tan x=\dfrac{\sin x}{\cos x}$ and $\sec x=\dfrac{1}{\cos x}$ in the given equation. Then, we will make the further calculations and multiply the square of cosx on both sides of the equation. Then, we will take ${{\sin }^{-1}}$ on both sides of the equation and again apply the trigonometric formula ${{\sin }^{-1}}\left( \sin x \right)=x$ in the equation and thus solve for x, which is our required result to the problem.

Complete step-by-step answer:
According to the question, we have to find the value of x from an equation.
Thus, we will use the trigonometric formulas to get the result.
The trigonometric equation given to us is $\tan x\sec x=0$ ------- (1)
So, we will first apply the trigonometric formula $\tan x=\dfrac{\sin x}{\cos x}$ and $\sec x=\dfrac{1}{\cos x}$ in equation (1), we get
$\dfrac{\sin x}{\cos x}.\dfrac{1}{\cos x}=0$
On further simplification, we get
$\dfrac{\sin x}{{{\cos }^{2}}x}=0$
Now, we will multiply the square of cosx on both sides of the equation, we get
$\dfrac{\sin x}{{{\cos }^{2}}x}.{{\cos }^{2}}x=0.{{\cos }^{2}}x$
As we know, the same terms in division cancel out each other with a quotient 1, therefore we get
$\sin x=0.{{\cos }^{2}}x$
Also, anything multiplied with number 0, is equal to 0, thus we get
$\sin x=0$
Now, we will put ${{\sin }^{-1}}$ on both sides in the above equation, we get
${{\sin }^{-1}}\left( \sin x \right)=~{{\sin }^{-1}}0$
Thus, we will apply the trigonometric formula ${{\sin }^{-1}}\left( \sin x \right)=x$ on the left-hand side of the above equation, we get
$x=~{{\sin }^{-1}}0$
As we know, the sin function is a periodic function, which means its value is the same after some interval, therefore we get
$x=~n\pi$ where n is some integer.
Therefore, for the trigonometric equation $\tan x\sec x=0$ , the value of x is equal to $n\pi$ where n is some integer.

Note: While solving this problem, do mention the trigonometric formula you are using in the steps. Do step-by-step calculation, to avoid confusion and mathematical error. At the final result, do not forget to write what n stands for in the answer.