Question

Step 1: startfraction sine (x minus y) over cosine (x minus y) endfraction step 2: startfraction sine (x) cosine (y) cosine (x) sine (y) over (a) cosine (y) minus sine (x) sine (y) endfraction step 3: startstartfraction startfraction sine (x) cosine (y) cosine (x) sine (y) over cosine (x) (b) endfraction overover startfraction cosine (x) cosine (y) minus sine (x) sine (y) over cosine (x) cosine (y) endfraction endendfraction step 4: startstartfraction startfraction sine (x) cosine (y) over cosine (x) cosine (y) endfraction startfraction cosine (x) sine over cosine (x) cosine (y) endfraction overover startfraction cosine (x) cosine (y) over cosine (x) cosine (y) endfraction minus startfraction sine (x) sine (y) over cosine (x) cosine (y) endfraction endendfraction step 5: the work shown is a way to derive tangent (x y) = startfraction tangent (x) tangent (y) over 1 minus tangent (x) tangent (y) endfraction what expressions go in the derivation of the tangent sum identity in place of a, b, and c? a: b:

Answer 1

In the derivation of the tangent sum identity, (a) should be replaced with 'cosine(x) cosine(y) + sine(x) sine(y)', (b) with '1 - sin²(x)', and (c) with 'sin(x)/cos(x)' which represent the cosine sum identity, the Pythagorean identity for cosine, and the definition of tangent respectively.

Step-by-step explanation:

The question refers to the derivation of the tangent sum identity, specifically tangent (x + y). The expressions that need to be inserted in place of (a), (b), and (c) in the given derivation stem from the fundamental trigonometric identities.

For (a), we use the identity cos(a - b) = cos a cos b + sin a sin b. For (b), we apply the identity cos²(x) = 1 - sin²(x), which is derived from the Pythagorean identity cos²(x) + sin²(x) = 1. The identity tan(x) = sin(x)/cos(x) is used to express tangent in terms of sine and cosine functions.

The steps in the derivation showcase these identities being used to rewrite the tangent of a sum of two angles in terms of the individual tangents of those angles.