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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

$alpha, beta in RR$ , prove that

$tan(alpha+beta) = (tan(alpha)+tan(beta))/(1-tan(alpha)tan(beta))$ . Assume that the addition formulas for sine and cosine are known.

	$\ \ \ \mathbf{ "Algebraic Steps" } \ \ \$	$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$	$\mathbf{ "Explanation" }$
	$(tan(alpha)+tan(beta))/(1-tan(alpha)tan(beta))$	$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$	$"(R.H.S. of the given equation)"$
		$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$
$1.$	$\ \ \ \ \ = (sin(alpha)/cos(alpha) + sin(beta)/cos(beta))/(1-sin(alpha)/cos(alpha) sin(beta)/cos(beta))$	$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$
		$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$
$2.$	$\ \ \ \ \ =$	$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$	$"Combine fractions in numerator"$
		$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$
$3.$	$\ \ \ \ \ = ((sin(alpha)cos(beta) \ + \ sin(beta)cos(alpha))/(cos(alpha)cos(beta)))/((cos(alpha)cos(beta))/(cos(alpha)cos(beta)) - (sin(alpha)sin(beta))/(cos(alpha)cos(beta)))$	$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$
		$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$
$4.$	$\ \ \ \ \ = ((sin(alpha)cos(beta) \ + \ sin(beta)cos(alpha))/(cos(alpha)cos(beta)))/((cos(alpha)cos(beta) \ - \ sin(alpha)sin(beta))/(cos(alpha)cos(beta)))$	$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$	$"Combine fractions in the denominator"$
		$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$
$5.$	$\ \ \ \ \ =$	$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$	$"Division of fractions"$
		$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$
$6.$	$\ \ \ \ \ = (sin(alpha)cos(beta) + sin(beta)cos(alpha))/(cos(alpha)cos(beta) - sin(alpha)sin(beta))$	$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$	$""$
		$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$
$7.$	$\ \ \ \ \ = sin(alpha+beta)/(cos(alpha)cos(beta) - sin(alpha)sin(beta))$	$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$	$""$
		$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$
$8.$	$\ \ \ \ \ = sin(alpha+beta)/cos(alpha+beta)$	$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$	$""$
		$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$
$9.$	$\ \ \ \ \ = tan(alpha+beta)$	$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$	$"Definition of tangent"$
		$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

Grade 11 Trigonometry CCSS: HSF-TF.C.9

What is the missing explanation in step 1?

$"Definition of tangent"$
$"Pythagorean Identity"$
$"Law of S""ines"$
$alpha/alpha = 1, \ beta/beta = 1$

Question Info

Trigonometry Question

Grade 11 Trigonometry CCSS: HSF-TF.C.9