Review the incomplete steps in the derivation of the tangent sum identity.

A 2-column table with 5 rows. Column 1 has entries Step 1: Original expression, step 2: rewrite using the definition of tangent, step 3: apply the sine sum identity and cosine sum identity, step 4: question mark, step 5: simplify using the definition of tangent. Column 2 has entries tangent (x + y), StartFraction sine (x + y) Over cosine (x + y) EndFraction, StartFraction sine (x) cosine (y) + cosine (x) sine (y) Over cosine (x) cosine (y) minus sine (x) sine (y) EndFraction, blank, StartFraction tangent (x) + tangent (y) Over 1 minus tangent (x) tangent (y) EndFraction.
Which action can be taken to the expression in Step 3 to give an expression for Step 4?
divide both the numerator and denominator by sin(x)sin(y)
divide both the numerator and denominator by sin(x)cos(y)
divide both the numerator and denominator by cos(x)sin(y)
divide both the numerator and denominator by cos(x)cos(y)

Answer:

D divide both the numerator and denominator by cos(x)cos(y)

Step-by-step explanation:

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The derivation of tangent sum identity taken to the expression in Step 3 to give an expression for Step 4 is option D; divide both the numerator and denominator by cos(x)cos(y).

The trigonometric identities is the relationship between the different trigonometric ratios.

These trigonometric identities are the basic formulae that are true for all the values of the reference angle.

For the given situation, the derivation for the tangent sum identity is given in the table below.

The derivation follows the steps,

[tex]\dfrac{sin (x+y)}{cos (x +y)}[/tex]

we know that,

sin (x + y) = sin x cos y + cos x sin y

cos (x + y) = cos x cos y - sin x sin y

Substitute;

[tex]\dfrac{sin x cos y + cos x sin y}{cos x cos y - sin x sin y}[/tex]

Divide each term by cos x cos y

[tex]\dfrac{\dfrac{sin x cos y}{cos x cos y} + \dfrac{cos x sin y}{cos x cos y} }{\dfrac{cos x cos y}{cos x cos y} - \dfrac{sin x sin y}{cos x cos y} }[/tex]

Now, [tex]\dfrac{tan x + tan y}{1 - tan x tan y}[/tex]

Hence we can conclude that the derivation of tangent sum identity taken to the expression in Step 3 to give an expression for Step 4 is option D; divide both the numerator and denominator by cos(x)cos(y).

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