/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 44 Verify the identity. $$\sec ^{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 44

Verify the identity. $$\sec ^{2} y-\cot ^{2}\left(\frac{\pi}{2}-y\right)=1$$

Short Answer

Expert verified
The given equation is valid and the identity is verified.

Step by step solution

01

Interpret the trigonometric functions

Firstly, Secant is the reciprocal of the Cosine function, i.e., \( \sec (y) = \frac{1}{\cos (y)} \). Also, Cotangent is the reciprocal of the Tangent function or the ratio of the cosine over sine i.e., \( \cot (x) = \frac{\cos (x)}{\sin (x)} \). We will use these identities and the property \( \sin^2(y) + \cos^2(y) = 1 \) in our exercise.
02

Change \( \cot \) function into \( \cos \) and \( \sin \)

Rewrite the equation as: \( \sec ^{2} y - \frac{\cos ^{2}\left(\frac{\pi}{2}-y\right)}{\sin ^{2} \left(\frac{\pi}{2} - y\right)}= 1\). Notice that we've taken \( \frac{\pi}{2}-y \) as 'x' for the identity of cotangent mentioned in Step 1.
03

Change \( \cos \left(\frac{\pi}{2}-y\right) \) and \( \sin \left(\frac{\pi}{2}-y\right) \) in terms of \( \cos \) and \( \sin \) of 'y'

For an angle \( \theta \), \( \cos \left(\frac{\pi}{2} - \theta\right) = \sin (\theta) \) and \( \sin \left(\frac{\pi}{2} - \theta\right) = \cos (\theta) \). Using these identities, we rewrite our equation as: \( \sec ^{2} y - \frac{\sin ^{2} y}{\cos ^{2} y} = 1 \).
04

Simplify the equation

We can rewrite the whole equation in terms of 'cos'. So, we get: \( \frac{1}{\cos ^{2} y} - \frac{\sin ^{2} y}{\cos ^{2} y} = 1 \). Furthermore, simplify to: \( \frac{1 - \sin ^{2} y}{\cos ^{2} y} = 1 \).
05

Utilize the trigonometric Pythagorean identity

We know that \( \sin ^{2} y + \cos ^{2} y = 1 \), so \( 1 - \sin ^{2} y = \cos ^{2} y \). Substituting in the equation we get: \( \frac{\cos ^{2} y}{\cos ^{2} y} = 1 \). This simplifies to 1 = 1, which confirms the identity.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the exact value of each expression. (a) \(\sin \left(135^{\circ}-30^{\circ}\right)\) (b) \(\sin 135^{\circ}-\cos 30^{\circ}\)

Solve the multiple-angle equation. $$\sec 4 x=2$$

A weight is oscillating on the end of a spring (see figure). The position of the weight relative to the point of equilibrium is given by \(y=\frac{1}{12}(\cos 8 t-3 \sin 8 t),\) where \(y\) is the displacement (in meters) and \(t\) is the time (in seconds). Find the times when the weight is at the point of equilibrium \((y=0)\) for \(0 \leq t \leq 1\).

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{17 \pi}{12}=\frac{9 \pi}{4}-\frac{5 \pi}{6}$$

Solve the multiple-angle equation. $$\cos 2 x=\frac{1}{2}$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.