Prove the identities provided.(tan(a+b))/(tan(a-b))=(sina cosa+sinb cosb)/(sina cosa-sinb cosb).

Name: Problem 48, Chapter 9: Algebra and Trigonometry
Uploaded: 2023-04-11T14:31:10-08:00
Duration: 8 min 11 s
Channel: Terrisa R
Description: Prove the identities provided.(tan(a+b))/(tan(a-b))=(sina cosa+sinb cosb)/(sina cosa-sinb cosb).

step 1 Okay we are going to verify the following identity. I have it all worked out here but I'm going

Prove the identities provided.(tan(a+b))/(tan(a-b))=(sina...

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold for all values in their domains. They provide the framework for manipulating and proving relationships between trigonometric expressions, which is essential for establishing the validity of any trigonometric equation.

Tangent as a Ratio of Sine and Cosine

The definition of tangent in terms of sine and cosine (tan x = sin x/cos x) is fundamental. This relationship allows one to express tangent-based expressions in terms of sine and cosine, facilitating comparisons and algebraic manipulation when proving identities.

Tangent Addition and Subtraction Formulas

The tangent addition (tan(a + b)) and subtraction (tan(a ? b)) formulas are key tools for breaking down complex tangent expressions into forms that use the tangent of individual angles. These formulas help transform the original expression into forms that reveal underlying patterns or simplifications.

Algebraic Manipulation in Trigonometric Proofs

Algebraic techniques such as factoring, combining fractions, and substitution are critical in the process of proving trigonometric identities. They allow one to reorganize and simplify expressions, ultimately demonstrating equivalence between different forms of a trigonometric expression.

Transcript

00:02 Okay we are going to verify the following identity.

00:04 I have it all worked out here but i'm going to step you through it as it's kind of complex.

00:10 So we're going to verify the tangent of a plus b divided by the tangent of a minus b equals sine a cosine a plus sine b cosine b divided by sine a cosine a minus sine b cosine b.

00:31 To do this we're going to use the sum and difference identity for the tangent.

00:38 So the tangent a plus b divided by the tangent a minus b.

00:43 We're going to do it on that.

00:44 We're going to work with the left hand side here and manipulate it to show that the left hand side is equal to the right.

00:50 You always work with one side and work your way down.

00:53 Okay so we're doing the left side and we use the sum and difference identity for tangent and that's going to become tangent a plus tangent b divided by one minus tangent a times tangent b divided by the quantity tangent a minus tangent b over one plus tangent a times tangent b.

01:11 Now remember you can take your numerator which is tangent a plus tangent b divided by one minus tangent a times tangent b times the reciprocal of your denominator which is one plus tangent a times tangent b divided by tangent a minus tangent b.

01:26 This makes it look a little nicer because now we're going to apply the tangent identity and if you remember that that tells us that the tangent of x equals the sine of x divided by cosine x.

01:47 So we are going to do that for every tangent that we have.

01:51 So we have sine a divided by cosine a.

01:54 Ignore this right now.

01:55 I'm going to erase this.

01:56 I'll show you what this is in a little bit here rather than confusing you yet.

02:04 So you can see we did sine a divided by cosine, and we put that in place of tangent a.

02:08 Sine b divided by cosine b in place of tangent b.

02:11 One minus sine a divided cosine a times sine b divided by cosine b and then we have one plus sine a divided by cosine a and then we have sine b divided by cosine b and sine a divided by cosine a minus sine b divided by cosine b and then we want to be able to add and subtract the numerators and denominators.

02:32 So to do that you want to get a least common denominator.

02:35 So in the numerator your least common denominator is cosine a times cosine b.

02:39 So i multiply the numerator and denominator by cosine b here and over here i'm going to multiply it by cosine a.

02:51 Then here you're going to multiply this entire quantity, the one, you're going to multiply it by cosine a times cosine b over cosine a times cosine b.

03:05 You would do the same thing on the right and that is so that you get common denominators you can add this.

03:12 So now i can do sine a cosine b plus sine b cosine a and it's all over the same denominator here of cosine a times cosine b.

03:25 Same thing here we get all over the same denominator of cosine a times cosine b.

03:33 Now if you remember you take the numerator times the reciprocal of the denominator which means these denominators cancel out...

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