Proofs of Tangent Addition & Subtraction Formulas (Grades 11-12)

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Proofs of Tangent Addition & Subtraction Formulas

1. 
For the following proof of the tangent addition formula, choose the correct missing expression or explanation for each applicable step in the questions below.


For [math]alpha, beta in RR[/math], prove that [math]tan(alpha+beta) = (tan(alpha)+tan(beta))/(1-tan(alpha)tan(beta))[/math]. Assume that the addition formulas for sine and cosine are known.

[math] [/math][math] \ \ \ \mathbf{ "Algebraic Steps" } \ \ \ [/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math]\mathbf{ "Explanation" }[/math]
[math][/math][math] (tan(alpha)+tan(beta))/(1-tan(alpha)tan(beta)) [/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math]"(R.H.S. of the given equation)"[/math]
[math][/math][math][/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math][/math]
[math]1.[/math][math] \ \ \ \ \ = (sin(alpha)/cos(alpha) + sin(beta)/cos(beta))/(1-sin(alpha)/cos(alpha) sin(beta)/cos(beta))[/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math][/math]
[math][/math][math][/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math][/math]
[math]2. [/math][math] \ \ \ \ \ = [/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math] "Combine fractions in numerator"[/math]
[math][/math][math][/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math][/math]
[math]3. [/math][math] \ \ \ \ \ = ((sin(alpha)cos(beta) \ + \ sin(beta)cos(alpha))/(cos(alpha)cos(beta)))/((cos(alpha)cos(beta))/(cos(alpha)cos(beta)) - (sin(alpha)sin(beta))/(cos(alpha)cos(beta)))[/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math][/math]
[math][/math][math][/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math][/math]
[math]4. [/math][math] \ \ \ \ \ = ((sin(alpha)cos(beta) \ + \ sin(beta)cos(alpha))/(cos(alpha)cos(beta)))/((cos(alpha)cos(beta) \ - \ sin(alpha)sin(beta))/(cos(alpha)cos(beta)))[/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math] "Combine fractions in the denominator"[/math]
[math][/math][math][/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math][/math]
[math]5. [/math][math] \ \ \ \ \ = [/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math] "Division of fractions"[/math]
[math][/math][math][/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math][/math]
[math]6. [/math][math] \ \ \ \ \ = (sin(alpha)cos(beta) + sin(beta)cos(alpha))/(cos(alpha)cos(beta) - sin(alpha)sin(beta))[/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math]""[/math]
[math][/math][math][/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math][/math]
[math]7. [/math][math] \ \ \ \ \ = sin(alpha+beta)/(cos(alpha)cos(beta) - sin(alpha)sin(beta))[/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math]""[/math]
[math][/math][math][/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math][/math]
[math]8. [/math][math] \ \ \ \ \ = sin(alpha+beta)/cos(alpha+beta)[/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math]""[/math]
[math][/math][math][/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math][/math]
[math]9. [/math][math] \ \ \ \ \ = tan(alpha+beta)[/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math] "Definition of tangent"[/math]
[math][/math][math][/math][math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math][math][/math]
A. 
What is the missing explanation in step 1?
  1. [math]"Definition of tangent"[/math]
  2. [math]"Pythagorean Identity"[/math]
  3. [math]"Law of S""ines"[/math]
  4. [math]alpha/alpha = 1, \ beta/beta = 1[/math]
B. 
What is the missing expression in step 2?
  1. [math](((sin(alpha)cos(beta))/(cos(alpha)sin(beta))))/((1-sin(alpha)/cos(alpha) sin(beta)/cos(beta)))[/math] [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math]
  2. [math](((sin(alpha)sin(beta))/(cos(alpha)cos(beta))))/((1-sin(alpha)/cos(alpha) sin(beta)/cos(beta)))[/math] [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math]
  3. [math](((sin(alpha)cos(alpha) \ + \ sin(beta)cos(beta))/(cos(alpha)cos(beta))))/((1-sin(alpha)/cos(alpha) sin(beta)/cos(beta)))[/math] [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math]
  4. [math](((sin(alpha)cos(beta) \ + \ sin(beta)cos(alpha))/(cos(alpha)cos(beta))))/((1-sin(alpha)/cos(alpha) sin(beta)/cos(beta)))[/math]
C. 
What is the missing explanation in step 3?
  1. [math]"The sine and cosine functions are never greater than 1"[/math]
  2. [math]cos^2(theta)+ sin^2(theta)=1[/math]
  3. [math]cos(0)=1, \ sin(90°)=1[/math]
  4. [math]"The number 1 can be rewritten as a given quantity divided by itself"[/math]
D. 
What is the missing expression in step 5?
  1. [math]( ( \ sin(alpha)cos(beta) + sin(beta)cos(alpha) \ ))/( ( \ cos(alpha)cos(beta) \ ) ( \ cos(alpha)cos(beta) - sin(alpha)sin(beta) \ ))[/math]
    [math] \ \ [/math]
  2. [math]( ( \ cos(alpha)cos(beta) \ )( \ sin(alpha)cos(beta) + sin(beta)cos(alpha) \ ))/(( \ cos(alpha)cos(beta) - sin(alpha)sin(beta) \ ))[/math]
    [math] \ \ [/math]
  3. [math]( ( \ cos(alpha)cos(beta) \ )( \ sin(alpha)cos(beta) + sin(beta)cos(alpha) \ ))/( ( \ cos(alpha)cos(beta) \ ) ( \ cos(alpha)cos(beta) - sin(alpha)sin(beta) \ ))[/math]
    [math] \ \ [/math]
  4. [math]( ( \ sin(alpha)cos(beta) \ )( \ sin(beta)cos(alpha) \ ))/( ( \ cos(alpha)cos(beta) \ ) ( \ sin(alpha)sin(beta) \ ))[/math]
    [math] \ \ [/math]
E. 
What is the missing explanation in step 6?
  1. Pythagorean Identity
  2. Cancel like factors
  3. Substitution Property of Equality
  4. Division Property of Identity
F. 
What is the missing explanation in step 7?
  1. Substitute in the sine addition formula
  2. Apply the Law of Sines
  3. Cancel like terms
  4. Factor out like terms
G. 
What is the missing explanation in step 8?
  1. Substitute in the cosine addition formula
  2. Apply the Law of Cosines
  3. Cancel like terms
  4. Factor out like terms
2. 
Given [math]tan(alpha + beta) = (tan(alpha)+tan(beta))/(1-tan(alpha)tan(beta))[/math] for [math]alpha, beta in RR[/math], prove that [math]tan(alpha-beta) = (tan(alpha)-tan(beta))/(1+tan(alpha)tan(beta))[/math].
A. 
Which of the following is equal to [math]tan(-x) ?[/math]
  1. [math]tan(x)[/math]
  2. [math]-tan(x)[/math]
  3. [math]-tan(-x)[/math]
  4. [math]tan(x-1)[/math]
B. 
Prove [math]tan(alpha-beta) = (tan(alpha)-tan(beta))/(1+tan(alpha)tan(beta))[/math], given that [math]tan(alpha + beta) = (tan(alpha)+tan(beta))/(1-tan(alpha)tan(beta))[/math].



3. 
Which of the following is equal to [math]tan(alpha-beta)/tan(alpha+beta) ?[/math]
  1. [math](sin(alpha)cos(alpha)) / (sin(beta)cos(beta))[/math]
  2. [math](sin(alpha)cos(beta) - sin(beta)cos(alpha)) / (sin(alpha)cos(alpha) + sin(beta)cos(beta))[/math]
  3. [math](sin(alpha)cos(alpha) - sin(beta)cos(beta)) / (sin(alpha)cos(alpha) + sin(beta)cos(beta))[/math]
  4. [math](sin(alpha)cos(alpha) - sin(beta)cos(beta)) / (sin(alpha)cos(beta) + sin(beta)cos(alpha))[/math]

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