What Is A Simplified Form Of The Expression $\frac{\sec ^2 X-1}{\sin X \sec X}$?A. $\cot X$ B. \$\csc X$[/tex\] C. $\tan X$ D. $\sec X \tan X$
**What is a Simplified Form of the Expression?** =====================================================
Introduction
In this article, we will explore the simplified form of the expression $\frac{\sec ^2 x-1}{\sin x \sec x}$. This expression involves trigonometric functions, and simplifying it will help us understand the relationships between these functions. We will use various trigonometric identities to simplify the expression and arrive at its final form.
Step 1: Simplify the Expression
To simplify the expression, we can start by using the identity $\sec ^2 x - 1 = \tan ^2 x$. This identity allows us to rewrite the numerator of the expression as $\tan ^2 x$.
\frac{\sec ^2 x-1}{\sin x \sec x} = \frac{\tan ^2 x}{\sin x \sec x}
Step 2: Simplify the Denominator
Next, we can simplify the denominator by using the identity $\sec x = \frac{1}{\cos x}$. This allows us to rewrite the denominator as $\frac{\sin x}{\cos x}$.
\frac{\tan ^2 x}{\sin x \sec x} = \frac{\tan ^2 x}{\sin x \frac{1}{\cos x}}
Step 3: Simplify the Expression Further
Now, we can simplify the expression further by canceling out the $\sin x$ terms in the numerator and denominator.
\frac{\tan ^2 x}{\sin x \frac{1}{\cos x}} = \frac{\tan ^2 x \cos x}{\sin x}
Step 4: Simplify the Expression Using Trigonometric Identities
Finally, we can simplify the expression using the identity $\tan x = \frac{\sin x}{\cos x}$. This allows us to rewrite the expression as $\frac{\tan ^2 x \cos x}{\sin x} = \tan x$.
\frac{\tan ^2 x \cos x}{\sin x} = \tan x
Conclusion
In conclusion, the simplified form of the expression $\frac{\sec ^2 x-1}{\sin x \sec x}$ is $\tan x$. This expression involves trigonometric functions, and simplifying it helps us understand the relationships between these functions.
Q&A
Q: What is the simplified form of the expression $\frac{\sec ^2 x-1}{\sin x \sec x}$? A: The simplified form of the expression is $\tan x$.
**Q: How do you simplify the expression $\frac\sec ^2 x-1}{\sin x \sec x}$?** A{\cos x}$.
**Q: What trigonometric identities are used to simplify the expression $\frac\sec ^2 x-1}{\sin x \sec x}$?** A{\cos x}$.
Q: What is the final form of the expression $\frac{\sec ^2 x-1}{\sin x \sec x}$? A: The final form of the expression is $\tan x$.
Q: How does simplifying the expression $\frac{\sec ^2 x-1}{\sin x \sec x}$ help us understand the relationships between trigonometric functions? A: Simplifying the expression helps us understand the relationships between trigonometric functions by showing how different functions can be expressed in terms of each other.
Q: What is the importance of simplifying trigonometric expressions? A: Simplifying trigonometric expressions is important because it helps us understand the relationships between different functions and can be used to solve problems in trigonometry and other areas of mathematics.