Log In Start studying!

Select your language

Suggested languages for you:

Americas

English (US)

Europe

Q72.

Expert-verified

Precalculus Mathematics for Calculus

Found in: Page 543

Book edition 7th Edition

Author(s) James Stewart, Lothar Redlin, Saleem Watson

Pages 948 pages

ISBN 9781337067508

Answers without the blur.

Just sign up for free and you're in.

Register for Free I’ll do it later

Short Answer

Proving Identities
Verify the identity $\frac{\cos^{2} t + \tan^{2} t - 1}{\sin^{2} t} = \tan^{2} t$

The expression $\frac{\cos^{2} t + \tan^{2} t - 1}{\sin^{2} t} = \tan^{2} t$ is an identity.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

An expression $\frac{\cos^{2} t + \tan^{2} t - 1}{\sin^{2} t} = \tan^{2} t$

Step 2. Concept used.

To demonstrate that the preceding statement is an identity, divide it into two portions, LHS and RHS. Separate the two and simplify them independently when you've completed separating. If the results are equal, the given statement is an identity.

Step 3. Calculation.

Now, simplify LHS of $\frac{\cos^{2} t + \tan^{2} t - 1}{\sin^{2} t} = \tan^{2} t$ :

Substituting 1 by $\cos^{2} t + \sin^{2} t$ ,

$\begin{array}{l} \frac{\cos^{2} t + \tan^{2} t - 1}{\sin^{2} t} = \frac{\cos^{2} t + \tan^{2} t - (\sin^{2} t + \cos^{2} t)}{\sin^{2} t} \\ = \frac{\tan^{2} t - \sin^{2} t}{\sin^{2} t} \\ (\tan x = \frac{\sin x}{\cos x}) \\ = \frac{\sin^{2} t}{\cos^{2} t (\sin^{2} t)} - \frac{\sin^{2} t}{\sin^{2} t} \end{array}$

$\begin{array}{l} \frac{\cos^{2} t + \tan^{2} t - 1}{\sin^{2} t} = \frac{1}{\cos^{2} t} - 1 \\ (\frac{1}{\cos x} = \sec x) \\ = \sec^{2} t - 1 \\ (\sec^{2} x + \tan^{2} x = 1) \\ = \tan^{2} t \end{array}$

RHS of the equation is $\tan^{2} t$

Both RHS and LHS are equal. Hence it is an identity

Most popular questions for Math Textbooks

Proving Identities

Verify the identity $\frac{1}{\sec x + \tan x} + \frac{1}{\sec x - \tan x} = 2 \sec x$

Simplifying Trigonometric Expressions-

Simplify the trigonometric expression $\frac{\sec x - \cos x}{\tan x}$

Simplifying Trigonometric Expressions $(\frac{\cos x \times \sec x}{\cot x})$

Simplifying Trigonometric Expressions-

Simplify the trigonometric expression $\frac{s e c^{2} x - 1}{s e c^{2} x}$

Proving Identities

Verify the identity $(1 - \cos β) (1 + \cos β) = \frac{1}{\csc^{2} β}$

Want to see more solutions like these?

Sign up for free to discover our expert answers

Get Started - It’s free

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free