Question

How do you simplify \[\tan x\csc x\]?

Answer 1

Hint: For solving this question, we should know that \[\tan x\] is equal to \[\dfrac{\sin x}{\cos x}\], \[\cot x\] is equal to the inverse of \[\tan x\], \[\sec x\] is equal to the inverse of \[\cos x\], \[\csc x\] is equal to the inverse of \[\sin x\]. We are going to use some of them to solve this problem.

Complete step by step answer:
Let us solve this question.
In this question, we have to simplify the term \[\tan x\csc x\] in simplest trigonometric form.
As we know that \[\tan x\] can be written as \[\dfrac{\sin x}{\cos x}\]
So, we can write the trigonometric term \[\tan x\csc x\] as the term \[\dfrac{\sin x}{\cos x}\csc x\].
And, also we know that \[\csc x\] can be written as the inverse of \[\sin x\] that is \[\dfrac{1}{\sin x}\].
So, the term \[\tan x\csc x\] which is equal to \[\dfrac{\sin x}{\cos x}\csc x\] can be written as \[\dfrac{\sin x}{\cos x}\dfrac{1}{\sin x}\].
Now, we can see that the term \[\sin x\] will be cancelled out as it is in both numerator and denominator.
So, the term \[\dfrac{\sin x}{\cos x}\dfrac{1}{\sin x}\] can be written as \[\dfrac{1}{\cos x}\].
Hence, we can say that \[\tan x\csc x\] can be written in the form of \[\dfrac{1}{\cos x}\].
And, we know that \[\dfrac{1}{\cos x}\] can be written as \[\sec x\].
Therefore, we can write
\[\tan x\csc x=\sec x\]

From here, we can say that the simplified form of \[\tan x\csc x\] can be written as \[\sec x\].

Note: We should remember the formulas, identities and equations of the trigonometry to solve this type of question. Formulas like \[\dfrac{1}{\cos x}=\sec x\], \[\dfrac{\sin x}{\cos x}=\tan x\], and \[\dfrac{1}{\sin x}=\csc x\], we have used in this question. So, don’t forget these types of formulas. They help a lot in solving questions.