Understand Relationship: Understand the relationship between secant and cosine. Secant is the reciprocal of cosine, so sec x = 1/cos x.Rewrite in Terms: Rewrite the expression in terms of cosine. (1)/(sec x - 1) can be rewritten as (1)/((1/cos x) - 1).Find Common Denominator: Find a common denominator to combine the terms in the denominator. The common denominator between 1/cos x and 1 is cos x. So, we rewrite the expression as (1)/((1 - cos x)/cos x).Multiply by Common Denominator: Multiply the numerator and the denominator by the common denominator (cos x) to simplify the expression. This gives us (cos x)/((1 - cos x)/cos x) * (cos x)/(cos x).Simplify Expression: Simplify the expression by canceling out the common factors. The cos x in the numerator and one of the cos x in the denominator cancel out, leaving us with (cos x)/(1 - cos x).

Algebra 2

Simplify rational expressions

Full solution

Q. b)

\frac{1}{\sec x - 1}

Understand Relationship: Understand the relationship between secant and cosine. $\newline$ Secant is the reciprocal of cosine, so $\sec x = \frac{1}{\cos x}.$
Rewrite in Terms: Rewrite the expression in terms of cosine. $\newline$ $(1)/(\sec x - 1)$ can be rewritten as $(1)/((1/\cos x) - 1)$ .
Find Common Denominator: Find a common denominator to combine the terms in the denominator. $\newline$ The common denominator between $\frac{1}{\cos x}$ and $1$ is $\cos x$ . So, we rewrite the expression as $\frac{1}{\left(\frac{1 - \cos x}{\cos x}\right)}$ .
Multiply by Common Denominator: Multiply the numerator and the denominator by the common denominator $\cos x$ to simplify the expression. $\newline$ This gives us $\frac{\cos x}{\left(\frac{1 - \cos x}{\cos x}\right)} \times \frac{\cos x}{\cos x}$ .
Simplify Expression: Simplify the expression by canceling out the common factors. $\newline$ The $\cos x$ in the numerator and one of the $\cos x$ in the denominator cancel out, leaving us with $\frac{\cos x}{1 - \cos x}$ .