(sec^(2)x-1)/(sec x-1)

Pythagorean Identity Recognition: We recognize that the numerator sec^(2)x - 1 is a Pythagorean identity which can be rewritten as tan^(2)x. sec^(2)x - 1 = tan^(2)xRewriting with Identity: Now we rewrite the original expression using the identity from step 1. (sec^(2)x - 1) / (sec x - 1) = (tan^(2)x) / (sec x - 1)Factoring Numerator: We factor the numerator tan^(2)x as tan x * tan x. (tan^(2)x) / (sec x - 1) = (tan x * tan x) / (sec x - 1)Substitution of Trigonometric Functions: We recognize that tan x is equal to sin x / cos x and sec x is equal to 1 / cos x. So we can rewrite tan x as sin x / cos x and sec x as 1 / cos x. (tan x * tan x) / (sec x - 1) = ((sin x / cos x) * (sin x / cos x)) / (1 / cos x - 1)Simplifying Fractions: We simplify the expression by multiplying both the numerator and the denominator by cos x to get rid of the fractions. ((sin x / cos x) * (sin x / cos x)) / (1 / cos x - 1) = (sin x * sin x) / (1 - cos x)Recognizing Trigonometric Identity: We recognize that sin x * sin x can be written as sin^(2)x. (sin x * sin x) / (1 - cos x) = sin^(2)x / (1 - cos x)Final Simplified Expression: We now have a simplified expression for the original problem. (sin^(2)x) / (1 - cos x)

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

$\left(\sec^{2}x-1\right)/\left(\sec x-1\right)$

View step-by-step help

Home

Math Problems

Algebra 2

Simplify radical expressions involving fractions

Full solution

\left(\sec^{2}x-1\right)/\left(\sec x-1\right)

Pythagorean Identity Recognition: We recognize that the numerator $\sec^2 x - 1$ is a Pythagorean identity which can be rewritten as $\tan^2 x$ . $\newline$ $\sec^2 x - 1 = \tan^2 x$
Rewriting with Identity: Now we rewrite the original expression using the identity from step $1$ . $\newline$ $(\sec^{2}x - 1) / (\sec x - 1) = (\tan^{2}x) / (\sec x - 1)$
Factoring Numerator: We factor the numerator $\tan^{2}x$ as $\tan x \cdot \tan x$ . $\newline$ $(\tan^{2}x) / (\sec x - 1) = (\tan x \cdot \tan x) / (\sec x - 1)$
Substitution of Trigonometric Functions: We recognize that $\tan x$ is equal to $\frac{\sin x}{\cos x}$ and $\sec x$ is equal to $\frac{1}{\cos x}$ . So we can rewrite $\tan x$ as $\frac{\sin x}{\cos x}$ and $\sec x$ as $\frac{1}{\cos x}$ . $\frac{\tan x \cdot \tan x}{\sec x - 1}$ = $\frac{(\frac{\sin x}{\cos x}) \cdot (\frac{\sin x}{\cos x})}{\frac{1}{\cos x} - 1}$
Simplifying Fractions: We simplify the expression by multiplying both the numerator and the denominator by $\cos x$ to get rid of the fractions. $\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) / \left(\frac{1}{\cos x} - 1\right) = \frac{\sin x \cdot \sin x}{1 - \cos x}$
Recognizing Trigonometric Identity: We recognize that $\sin x \cdot \sin x$ can be written as $\sin^{2}x$ . $\newline$ $(\sin x \cdot \sin x) / (1 - \cos x) = \sin^{2}x / (1 - \cos x)$
Final Simplified Expression: We now have a simplified expression for the original problem. $\frac{\sin^{2}x}{1 - \cos x}$