Establish each identity. (sin^3 θ+cos^3 θ)/(1-2 cos^2 θ)=(secθ-sinθ)/(tanθ-1)

Name: Problem 84, Chapter 8: Algebra and Trigonometry
Uploaded: 2020-06-20T13:41:07-08:00
Duration: 8 min 4 s
Channel: Peter Winans
Description: Establish each identity. (sin^3 θ+cos^3 θ)/(1-2 cos^2 θ)=(secθ-sinθ)/(tanθ-1)

step 1 All right. We want to prove that this identity is true. We're going to show that the left side a

Establish each identity. (sin^3 θ+cos^3 θ)/(1-2 cos^2...

Key Concepts

Sum of Cubes Factorization

This concept involves factoring expressions of the form a³ + b³ into the product (a + b)(a² - ab + b²). In the context of trigonometric functions, when expressions like sin³? + cos³? appear, applying this factorization simplifies the expression and aids in further cancellation or transformation steps necessary to establish the identity.

Pythagorean Identity

The Pythagorean identity, sin²? + cos²? = 1, is fundamental in trigonometry and is used to simplify expressions by replacing combinations of sin²? and cos²? with 1. This simplification is pivotal when reducing complex trigonometric expressions to more manageable forms during the verification of identities.

Reciprocal and Quotient Identities

Reciprocal identities (like sec? = 1/cos?) and quotient identities (like tan? = sin?/cos?) convert trigonometric expressions into equivalent forms that are easier to manipulate algebraically. These identities bridge the gap between different trigonometric functions, enabling the re-expression and cancellation of terms to confirm the given identity.

Algebraic Manipulation of Trigonometric Expressions

This concept encompasses the techniques used to simplify and prove trigonometric identities, including factorization, common denominator extraction, and cancellation of common terms. Such algebraic manipulations, in combination with the trigonometric identities, allow for the rearrangement and transformation of the expression into an equivalent form, thereby establishing the validity of the identity.

Transcript

00:03 All right.

00:04 We want to prove that this identity is true.

00:07 We're going to show that the left side and the right side are the same.

00:10 And this is going to be a complicated one.

00:12 It's going to be a little bit long, but that's okay.

00:16 The first thing that we're going to do is i'm looking at the sum of two perfect cubes in my numerator.

00:24 And i think we're going to use the formula to factor those.

00:28 So when i factor sine cube plus cosine cubed, i'm going to get sine of theta plus the number.

00:36 Cosine of theta times and then we're going to get sine squared of theta minus sine of theta cosine of theta plus cosine squared of theta and all of that is going to be over one minus two cosine squared theta.

01:13 Now it's not immediately obvious what we have to do next here but we can do some playing around.

01:26 So the next thing that i'm going to do is recognize that in this second term, i have sine squared plus cosine squared.

01:41 And if it makes us feel better, we can rearrange our terms so that they're next to each other.

01:46 In fact, maybe we'll just do that.

01:48 We're going to have sine theta plus cosine of theta times sine square theta plus cosine square of theta minus sine theta cosine of theta and i'm going to need that all to be over i'm looking at this i'm kind of looking ahead here a little bit as a clue just to explain why i'm going to do what i'm about to do in my denominator i have the sine theta plus cosine of theta here and i might have a chance at reducing that if i can get some other factors here so in my denominator i am going to take the negative 2 cosine square theta, and i'm going to write that as minus cosine square theta minus cosine square theta, right? negative cosine squared, minus cosine squared is going to be negative two cosine squared.

02:56 So i haven't really done anything there.

02:58 I just kind of split that up into two terms.

03:00 The reason i like that is because i'm going to be able to use the pythagorean theorem or pythagorean identity twice now in different forms.

03:09 The first, sine squared plus cosine squared is equal to 1.

03:13 And the second, in our denominator, a different form of the pythagorean identity states that sine squared would be equal to 1 minus cosine squared.

03:22 So i'm going to get sine of theta plus the cosine of theta times we're going to have, we're going to have one.

03:42 I'm going to make that substitution minus the sine of theta times the cosine of theta.

03:51 And this now is going to be all over, sine square theta when i make that substitution, minus the cosine squared of theta.

04:04 Now, the reason i like that, the reason i did that is because i have, again, the sine theta plus cosine theta there, i might be able to reduce it.

04:13 So now in my denominator, i'm looking at the difference of two squares.

04:21 So we will have, as our next step, sine theta plus the cosine of theta times one minus the sine of theta times the cosine of theta all over...

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