Question

How do I simplify $\dfrac{{\sec x}}{{\tan x}}$?

Answer 1

Here, we will simplify the trigonometric expression by using the trigonometric ratio and trigonometric reciprocal function in the given expression to get the required answer. Trigonometric ratios are used to find the relationships between the sides of a right-angle triangle.

Formula Used:
We will use the following formulas:
Trigonometric reciprocal function $\sec x = \dfrac{1}{{\cos x}}$
Trigonometric Ratio $\tan x = \dfrac{{\sin x}}{{\cos x}}$
Trigonometric reciprocal function $\dfrac{1}{{\sin x}} = \cos ecx$

Complete Step by Step Solution:
We are given a trigonometric expression $\dfrac{{\sec x}}{{\tan x}}$.
Let $f\left( x \right)$ be the given trigonometric equation. Thus, we get
$f\left( x \right) = \dfrac{{\sec x}}{{\tan x}}$
We know that Trigonometric reciprocal function $\sec x = \dfrac{1}{{\cos x}}$ and Trigonometric Ratio $\tan x = \dfrac{{\sin x}}{{\cos x}}$
Now, we will rewrite the given trigonometric expression in terms of sine and cosine.
By using the Trigonometric reciprocal function and Trigonometric Ratio in the given Trigonometric Expression, we get
$\Rightarrow f\left( x \right) = \dfrac{{\dfrac{1}{{\cos x}}}}{{\dfrac{{\sin x}}{{\cos x}}}}$
By rewriting the expression, we get
$\Rightarrow f\left( x \right) = \dfrac{1}{{\cos x}} \cdot \dfrac{{\cos x}}{{\sin x}}$
By canceling out the like terms, we get
$\Rightarrow f\left( x \right) = \dfrac{1}{{\sin x}}$
We know that Trigonometric reciprocal function $\dfrac{1}{{\sin x}} = \cos ecx$
By using the Trigonometric reciprocal function, we get
$\Rightarrow f\left( x \right) = \cos ecx$

Therefore, the given trigonometric expression $\dfrac{{\sec x}}{{\tan x}}$ is $\cos ecx$.

Note: We know that the Trigonometric equation is defined as an equation involving trigonometric ratios. Trigonometric identity is an equation that is always true for all the variables. There are many trigonometric identities that are related to all the other trigonometric equations. We should also remember that the trigonometric ratio and the co-trigonometric ratio is always reciprocal to each other. Trigonometric ratios of a Particular angle are the ratios of the sides of a right-angled triangle with respect to any of its acute angles.