### Proving the Identity in TrigonometryWe are given the following trigonometric identity to prove:\[ \cot(x - y) = \frac{\cot(x) \cdot \cot(y) + 1}{\cot(y) - \cot(x)} \]We'll prove this identity by using reciprocal identities and subtraction formulas step by step.#### Step 1: Use a Reciprocal Identity, and then use a Subtraction Formula.Starting with the left-hand side:\[ \cot(x - y) = \frac{1}{\tan(x - y)} \]Using the tangent subtraction formula:\[ \tan(x - y) = \frac{\tan(x) - \tan(y)}{1 + \tan(x) \cdot \tan(y)}\]So by reciprocal identity:\[ \cot(x - y) = \frac{1}{\frac{\tan(x) - \tan(y)}{1 + \tan(x) \cdot \tan(y)}} \]\[ = \frac{1 + \tan(x) \cdot \tan(y)}{\tan(x) - \tan(y)} \]#### Step 2: Use a Reciprocal Identity, and simplify the compound fraction.Next, express tangents in terms of cotangents:\[ \tan(x) = \frac{1}{\cot(x)}, \quad \tan(y) = \frac{1}{\cot(y)} \]Substituting these into the equation:\[ \cot(x - y) = \frac{1 + \frac{1}{\cot(x)} \cdot \frac{1}{\cot(y)}}{\frac{1}{\cot(x)} - \frac{1}{\cot(y)}} \]Simplifying the numerator and the denominator:\[ = \frac{1 + \frac{1}{\cot(x) \cdot \cot(y)}}{\frac{1}{\cot(x)} - \frac{1}{\cot(y)}} \]\[ = \frac{1 + \frac{1}{\cot(x) \cot(y)}}{\frac{\cot(y) - \cot(x)}{\cot(x) \cdot \cot(y)}} \]Further simplify to:\[ = \frac{(1 + \frac{1}{\cot(x) \cdot \cot(y)}) \cdot (\cot(x) \cdot \cot(y))}{ **Verify the Identity**\[ \frac{1}{\sec(x) + \tan(x)} + \frac{1}{\sec(x) - \tan(x)} = 2 \sec(x) \]To verify this trigonometric identity, we start by considering the left-hand side:\[ \frac{1}{\sec(x) + \tan(x)} + \frac{1}{\sec(x) - \tan(x)} \]First, find a common denominator:\[ \frac{\sec(x) - \tan(x)}{(\sec(x) + \tan(x))(\sec(x) - \tan(x))} + \frac{\sec(x) + \tan(x)}{(\sec(x) + \tan(x))(\sec(x) - \tan(x))} \]Combine the fractions:\[ \frac{\sec(x) - \tan(x) + \sec(x) + \tan(x)}{(\sec(x) + \tan(x))(\sec(x) - \tan(x))} \]Simplify the numerator:\[ \frac{\sec(x) + \sec(x)}{\sec^2(x) - \tan^2(x)} \]\[ \frac{2 \sec(x)}{\sec^2(x) - \tan^2(x)} \]Noting that:\[ \sec^2(x) - \tan^2(x) = 1 \]The expression becomes:\[ \frac{2 \sec(x)}{1} = 2 \sec(x) \]Thus, we have shown that:\[ \frac{1}{\sec(x) + \tan(x)} + \frac{1}{\sec(x) - \tan(x)} = 2 \sec(x) \]This verifies the given identity.

Name: ### Proving the Identity in TrigonometryWe are given the following tr
Uploaded: 2024-03-20T06:46:42.000Z
Channel: Bartleby
Description: ### Proving the Identity in TrigonometryWe are given the following trigonometric identity to prove:\[ \cot(x - y) = \frac{\cot(x) \cdot \cot(y) + 1}{\cot(y) - \cot(x)} \]We'll prove this identity by using reciprocal identities and subtraction formul

Answered: Image /qna-images/answer/aaf2f727-0286-4e44-a403-dea2cf178d26.jpg

Answered: cot(x) cot(y) + 1 cot(y) – cot(x) cot(x…

Math

Trigonometry

cot(x) cot(y) + 1 cot(y) – cot(x) cot(x – y) =

Trigonometry (11th Edition)

11th Edition

ISBN:9780134217437

Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Chapter1: Trigonometric Functions

Section: Chapter Questions

Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.

See similar textbooks

Related questions

Q: please find the first and second derivatives and show complete solution

Q: sec x Csc x tan x + cot x tan x cot x sec x + cS x -

A: Given that The trigonometric ration To show that lhs=rhs

Q: 6tanx- COSX 2 - 3 sin x

Q: draw a graph of the function using to the informalion from coordinales Obtained the first and…

Q: 4.空+= 5(x-2)、 dx ズー2

A: Explanation of the answer is as follows

Q: flæ)dr

A: SOLUTION-

Q: 4. -5x2

A: As per our guidelines we are supposed to answer only one question.Kindly repost other question as…

Q: Cosx Find (x), f (x) +{In(Tan). 2Sin?x 2 ||

Q: 3x" -2x+5 1. 4x2-3х+5 2. 2сos(30?) |2у+4y 3. V y²-4

Q: From curve equation y=x²-2x+1 at (2,1) Find?

Q: From curve equation y=x²-2x +1 at (2,1) Find ?

Q: 4 sec*(x) rind tha ualı Find the value of ((x* - 5x² – 6x)“ 4x³ - 10x – 6 dx. (x4–5x²–6x)* - a) 4…

Q: 6x2 y = 2y + cos y

A: We need to find the derivative of the given function

Q: fy cot-2(x

Q: show your complete solution Find the first, second and third derivative of y = 5 cos(x) + (x + 5)-2…

Q: Find the derivative of the trigonometic function, no chain rule please! 2.2 & 2.3 rules only…

Q: Use a graphing utility to graph each function in the Interval (0, 27). y2 cos(x), y 2x + 2 (a) write…

A: Given y=2cos(x)y=2x+2x2interval [0,2π)

Q: Question 3 Let f(z) 2r-3z2-6z+2. Find the the equations of all tangent lines to f which are parallel…

Q: 1.) f (x) = 10x10 4.) f (x) = %3D 2.) f(x) = x² – 9 5.) f(x) = 2x² + Vx 3.) f (x) = –x5 – x-3 – - x

Q: QUESTION 8 If then the value of y' is 3cosx - 10 3sinx - 10 3cosx + 10

Q: 2) Inverse of function, y =;+1, with check 5

A: As this is a multiple question according to the Bartleby Answering rule, only first question is to…

Q: Question 3 Find two functions f(x) and g(x) so that h(x) = (fog)(x) a) h(x) = (2x +3)². %3D

A: The function h(x) is defined as hx=fοgx

Q: Given that - ≤ß<, which of the following gives all solutions to the equation Select the correct…

A: Formula used 1 + tan2β = sec2βsec2β - 1 = tan2βtan( -β ) = -tanβGiven sec2β - 1tan( -β ) = 1

Q: 4 2х 3. Derive y = 4х-3

A: Follow next step

Q: Find the direction 0 in which the directional derivative of the function f(x, y) = æy+ y² at the…

Q: QUESTION 1 What is y" if y = x2 when x = 2? О а. -3/8 O b. 1/4 О с. 3/8 O d.-1/4

A: Given : y=x-2 We need to find y'' at x=2.

Q: Let z = CSC y y³ Then dz dy csc (y) — csc (y) [1³] X

Q: 3) Given y = sin x. State the domain and range (#2,1) (x.0) (2x, 0). (0,0) (3x2-1) Domain = Range =…

A: We have given y=sinx Domain is the values of x for which a function is defined. y=sinx is defined…

Topic Video

Question

100%

### Proving the Identity in Trigonometry

We are given the following trigonometric identity to prove:

\[ \cot(x - y) = \frac{\cot(x) \cdot \cot(y) + 1}{\cot(y) - \cot(x)} \]

We'll prove this identity by using reciprocal identities and subtraction formulas step by step.

#### Step 1: Use a Reciprocal Identity, and then use a Subtraction Formula.

Starting with the left-hand side:
\[ \cot(x - y) = \frac{1}{\tan(x - y)} \]

Using the tangent subtraction formula:
\[ \tan(x - y) = \frac{\tan(x) - \tan(y)}{1 + \tan(x) \cdot \tan(y)}\]

So by reciprocal identity:
\[ \cot(x - y) = \frac{1}{\frac{\tan(x) - \tan(y)}{1 + \tan(x) \cdot \tan(y)}} \]
\[ = \frac{1 + \tan(x) \cdot \tan(y)}{\tan(x) - \tan(y)} \]

#### Step 2: Use a Reciprocal Identity, and simplify the compound fraction.

Next, express tangents in terms of cotangents:
\[ \tan(x) = \frac{1}{\cot(x)}, \quad \tan(y) = \frac{1}{\cot(y)} \]

Substituting these into the equation:
\[ \cot(x - y) = \frac{1 + \frac{1}{\cot(x)} \cdot \frac{1}{\cot(y)}}{\frac{1}{\cot(x)} - \frac{1}{\cot(y)}} \]

Simplifying the numerator and the denominator:
\[ = \frac{1 + \frac{1}{\cot(x) \cdot \cot(y)}}{\frac{1}{\cot(x)} - \frac{1}{\cot(y)}} \]
\[ = \frac{1 + \frac{1}{\cot(x) \cot(y)}}{\frac{\cot(y) - \cot(x)}{\cot(x) \cdot \cot(y)}} \]

Further simplify to:
\[ = \frac{(1 + \frac{1}{\cot(x) \cdot \cot(y)}) \cdot (\cot(x) \cdot \cot(y))}{

**Verify the Identity**

\[ \frac{1}{\sec(x) + \tan(x)} + \frac{1}{\sec(x) - \tan(x)} = 2 \sec(x) \]

To verify this trigonometric identity, we start by considering the left-hand side:

\[ \frac{1}{\sec(x) + \tan(x)} + \frac{1}{\sec(x) - \tan(x)} \]

First, find a common denominator:

\[ \frac{\sec(x) - \tan(x)}{(\sec(x) + \tan(x))(\sec(x) - \tan(x))} + \frac{\sec(x) + \tan(x)}{(\sec(x) + \tan(x))(\sec(x) - \tan(x))} \]

Combine the fractions:

\[ \frac{\sec(x) - \tan(x) + \sec(x) + \tan(x)}{(\sec(x) + \tan(x))(\sec(x) - \tan(x))} \]

Simplify the numerator:

\[ \frac{\sec(x) + \sec(x)}{\sec^2(x) - \tan^2(x)} \]

\[ \frac{2 \sec(x)}{\sec^2(x) - \tan^2(x)} \]

Noting that:

\[ \sec^2(x) - \tan^2(x) = 1 \]

The expression becomes:

\[ \frac{2 \sec(x)}{1} = 2 \sec(x) \]

Thus, we have shown that:

\[ \frac{1}{\sec(x) + \tan(x)} + \frac{1}{\sec(x) - \tan(x)} = 2 \sec(x) \]

This verifies the given identity.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Chain Rule$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, trigonometry and related others by exploring similar questions and additional content below.

Recommended textbooks for you

Trigonometry (11th Edition)

Trigonometry

ISBN:

9780134217437

Author:

Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:

PEARSON