Question

How do you simplify \[\dfrac{1}{{\sin x}}\]?

Answer 1

Hint: In the given question, we have been given a trigonometric expression. We have to simplify this expression. This expression involves a single trigonometric function in the reciprocal. So, we just have to write the trigonometric function which is the reciprocal of \[\sin x\].

Formula Used:
We are going to use the formula:
\[\sin x \times \csc x = 1\]

Complete step-by-step answer:
The expression to be simplified is
\[\dfrac{1}{{\sin x}}\]
So, we just have to find the trigonometric expression which is the reciprocal of \[\sin x\].
To do that, we write all the known trigonometric functions:
\[\sin x,\cos x,\sec x,{\mathop{\rm cosec}\nolimits} x,\cot x,\tan x\]
We know,
\[\tan x = \dfrac{1}{{\cot x}}\]
and \[\cos x = \dfrac{1}{{\sec x}}\]
Hence, the only thing left is
\[\sin x = \dfrac{1}{{{\mathop{\rm cosec}\nolimits} x}}\]
or \[{\mathop{\rm cosec}\nolimits} x = \dfrac{1}{{\sin x}}\]
Hence, \[\dfrac{1}{{\sin x}} = {\mathop{\rm cosec}\nolimits} x\]

Note: In the given question, we had been given a trigonometric expression. We had to simplify this expression. This expression had a single trigonometric function in the denominator. So, we just had to think of the trigonometric function which is the reciprocal of the given trigonometric expression. We just wrote all the trigonometric functions and picked the one.