Use trig identities to simplify an expression (has sins and cosines)

Click For Summary
SUMMARY

The discussion focuses on simplifying the trigonometric expression \((\sin x)^2 - (\cos x)^2\) divided by \((\sin x)^2 - (\sin x \cos x)\). The solution involves recognizing that the numerator can be factored as a difference of squares, leading to the simplification of the expression to \(1 + \cot x\). Key identities used include \((\sin x)^2 + (\cos x)^2 = 1\) and the manipulation of the expression by multiplying both the numerator and denominator by \(1/\sin^2(x)\) to facilitate the simplification.

PREREQUISITES
  • Understanding of fundamental trigonometric identities
  • Ability to factor expressions, particularly differences of squares
  • Familiarity with the cotangent function and its relationship to sine and cosine
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the derivation and applications of the Pythagorean identity \((\sin x)^2 + (\cos x)^2 = 1\)
  • Learn how to factor expressions using the difference of squares method
  • Explore the properties and applications of the cotangent function in trigonometry
  • Practice simplifying complex trigonometric expressions using various identities
USEFUL FOR

Students studying trigonometry, educators teaching trigonometric identities, and anyone looking to enhance their skills in simplifying trigonometric expressions.

Nishiura_high
Messages
7
Reaction score
0

Homework Statement



Use fundamental identities to simplify the expression:

(sinx)^2 - (cosx)^2
____________________
(sinx)^2 - (sinx cosx)*note: it's a numerator and denominator. The underscore line is the fraction line.

*note: The answer in the back of the book is "1 + cotx" but I would like to know how it got there.

Homework Equations



(sinx)^2 + (cosx)^2 = 1

other trig identities

The Attempt at a Solution



(sinx)^2 - (cosx)^2
_________________
sinx(sinx - cosx)

I factored out sinx out of the bottom, but I don't really see any identies that would simplify sinx-cosx. (I have a chart of identities.) I tried to simplify the top using the relevant identity I already listed.

Thanks for any help!
 
Last edited:
Physics news on Phys.org
Nishiura_high said:

Homework Statement



Use fundamental identities to simplify the expression:

(sinx)^2 - (cosx)^2
____________________
(sinx)^2 - (sinx cosx)

*note: it's a numerator and denominator. The underscore line is the fraction line.

*note: The answer in the back of the book is "1 + cotx" but I would like to know how it got there.

Homework Equations



(sinx)^2 + (cosx)^2 = 1

other trig identities


The Attempt at a Solution



(sinx)^2 - (cosx)^2
_________________
sinx(sinx - cosx)

I factored out sinx out of the bottom, but I don't really see any identies that would simplify sinx-cosx. (I have a chart of identities.) I tried to simplify the top using the relevant identity I already listed.

Thanks for any help!
Factor the numerator as a difference of squares.

or ...

Starting with the original expression, multiply the numerator and denominator by 1/sin2(x)
 
Thanks. I got it now. :)
 

Similar threads

Need help proving some trig identities
  • · Replies 4 ·
Replies
4
Views
2K
Solving Harder Trig Identity - Peter's Question
  • · Replies 8 ·
Replies
8
Views
2K
Verify the Trigonometric Identity
  • · Replies 15 ·
Replies
15
Views
3K
How to Solve Trig Identity Questions: Tips and Examples
  • · Replies 28 ·
Replies
28
Views
5K
Finding points of intersection algebraically between 2 trig functions
  • · Replies 11 ·
Replies
11
Views
36K
Solve Simple Trig Problem: cosx - cosx*sinx = 1/3
  • · Replies 2 ·
Replies
2
Views
2K
Trig identities (is my method correct?)
  • · Replies 6 ·
Replies
6
Views
3K
Trig ((tanx - sinx)/sin^3x) = (1/(cosx - cos^2x))
  • · Replies 4 ·
Replies
4
Views
7K
How Can You Prove This Trigonometric Identity?
  • · Replies 2 ·
Replies
2
Views
4K
Unsure if this is a trig identity or calculus
  • · Replies 7 ·
Replies
7
Views
2K