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Question: How do you solve \(\sin x\tan x-\sin x=0\) ?...

How do you solve sin⁡xtan⁡x−sin⁡x=0\sin x\tan x-\sin x=0 ?

Explanation

Solution

In this question, we have to find the value of x of a trigonometric equation. Therefore, we will use the trigonometric formulas to get the solution. We will first take sinx common from the given equation. After that, we will solve two separate equations, to get the required solution for the problem.

Complete step-by-step answer:
According to the problem, we have to find the value of x from an equation.
We will use the trigonometric formula to get the required result for the problem.
The trigonometric equation given to us is sin⁡xtan⁡x−sin⁡x=0\sin x\tan x-\sin x=0 -------- (1)
We will first take sinx common from equation (1), we get
sin⁡x(tan⁡x−1)=0\sin x(\tan x-1)=0
Therefore, we get two separate equations from the above equation, we get
sin⁡x=0\sin x=0 and -------- (2)
(tan⁡x−1)=0(\tan x-1)=0 --------- (3)
Now, we will solve equation (2), which is
sin⁡x=0\sin x=0
So, take the inverse of sin function on both sides in the above equation, we get
sin⁡−1(sin⁡x)=sin⁡−10{{\sin }^{-1}}\left( \sin x \right)={{\sin }^{-1}}0
So, we will apply the trigonometric formula sin⁡−1(sin⁡x)=x{{\sin }^{-1}}\left( \sin x \right)=x in the above equation, we get
x=sin⁡−10x={{\sin }^{-1}}0
Therefore, on further simplification, we get
x=nπx=n\pi where n is some integer.
Now, we will solve equation (3), which is
tan⁡x−1=0\tan x-1=0
Now, we will add 1 on both sides of the equation, we get
tan⁡x−1+1=0+1\tan x-1+1=0+1
As we know, the same terms with opposite signs cancel out each other, therefore we get
tan⁡x=1\tan x=1
So, take the inverse of tab function on both sides in the above equation, we get
tan⁡−1(tan⁡x)=tan⁡−11{{\tan }^{-1}}\left( \tan x \right)={{\tan }^{-1}}1
So, we will apply the trigonometric formula tan⁡−1(tan⁡x)=x{{\tan }^{-1}}\left( \tan x \right)=x in the above equation, we get
x=tan⁡−11x={{\tan }^{-1}}1
Therefore, on further simplification, we get
x=π4+nπx=\dfrac{\pi }{4}+n\pi where n is some integer.
Therefore, for the trigonometric equation sin⁡xtan⁡x−sin⁡x=0\sin x\tan x-\sin x=0 , we get two values of x that is
x=nπ,π4+nπx=n\pi ,\dfrac{\pi }{4}+n\pi where n is some integer.

Note: While solving this problem, keep in mind the formula you are using to solve the problem. Do step-by-step calculations to avoid confusion and mathematical error. One of the alternative methods to solve this problem is to convert the tan function into sin and cos function. Then, we will take the sin function common and make a separate equation to solve for x.