Answer

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**Hint:**

Here we will find the value of the given trigonometric function. First, we will convert the given trigonometric function into sine function by using the reciprocal trigonometric identity. Then we will use the calculator to find the value of obtained sine function. Then we will simplify the equation further to get the required answer.

**Complete step by step solution:**

We have to find the value of \[\csc 10\].

We know that cosecant is also define as reciprocal sine function i.e. \[\csc x = \dfrac{1}{{\sin x}}\].

Therefore, we will convert the given cosecant function in sine function as,

\[\csc 10 = \dfrac{1}{{\sin 10}}\]…..\[\left( 1 \right)\]

Next, we will use calculator and find the value of \[\sin 10\] and substitute it in above equation. Therefore, we get

\[ \Rightarrow \csc 10 = \dfrac{1}{{ - 0.54402}}\]

Dividing the terms, we get

\[ \Rightarrow \csc 10 = - 1.838\]

**So we get the value of \[\csc 10\] as \[ - 1.838\].**

**Additional information:**

The Reciprocal Identity of trigonometric functions state that there are three functions which can be defined as the reciprocal of the other three functions such as, secant can be defined as reciprocal of cosine, cosecant can be defined as reciprocal of sine and cotangent can be defined as reciprocal of the tangent.

**Note:**

Trigonometry is that branch of mathematics that deals with specific functions of angles and also their application in calculations and simplification. The commonly used six types of trigonometry functions are defined as sine, cosine, tangent, cotangent, secant and cosecant. Identities are those equations which are true for every variable. The trigonometric functions are those real functions that relate the angle to the ratio of two sides of a right-angled triangle.

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