Question

How do you simplify the expression \[\dfrac{1}{{\tan x}}\]?

Answer 1

Hint: In the given question, we have been given an expression. The expression is a trigonometric expression. This trigonometric expression consists of just one trigonometric function. And this trigonometric function is in the denominator. And the numerator corresponding to this trigonometric function is One. And there are no other trigonometric functions or anything else. We have to simplify this trigonometric expression and express it in the form of other trigonometric functions. For solving that, we only need to know which is that trigonometric function whose value for the same angle of this trigonometric function, is its reciprocal.

Complete step-by-step answer:
The given expression is \[\dfrac{1}{{\tan x}}\].
The given trigonometric function is \[\tan x\]. We have to find the trigonometric function whose product with \[\tan x\] gives us \[1\].
Now, \[\tan x = \dfrac{{\sin x}}{{\cos x}}\]
So, we need to find the trigonometric function which is equal to
\[\dfrac{{\cos x}}{{\sin x}}\]
And we know, the trigonometric function which is equal to this is \[\cot x\].
Hence, \[\dfrac{1}{{\tan x}} = \cot x\]

Additional Information:
In the question, we had to calculate the reciprocal of \[\tan x\], which is \[\cot x\], and the reverse is also true. But if we were given \[\sin x\], then its reciprocal, hence the answer, would have been \[{\mathop{\rm cosec}\nolimits} x\].
And similarly, for the same thing for \[\cos x\], it would have been \[\sec x\].

Note: In the given question, we just had to know what is that trigonometric function which when multiplied with the trigonometric function in the question, always gives the answer one, or, what is the reciprocal of the given trigonometric function.