Question

Proof of cosine half-angle identity: 1. cos(2θ) = 2 cos2θ – 1 given 2. 1 cos(2θ) = 2cos2θ add 1 to both sides 3. divide by 2 4. cosine squared (startfraction theta over 2 endfraction) = startstartfraction 1 cosine (2 (startfraction theta over 2 endfraction) ) overover 2 endfraction blank line. ________ 5. cosine squared (startfraction theta over 2 endfraction) = startfraction 1 cosine theta over 2 endfraction multiplication 6. cosine (startfraction theta over 2 endfraction) = plus-or-minus startroot startfraction 1 cosine theta over 2 endfraction endroot square root analyze the proof. use the drop-down boxes to complete the sentences. the third line of this proof is known as the . identity of cosine. the justification for the fourth line is .

Answer 1

The third line of the proof which involves dividing by 2 is where the half-angle identity is derived, establishing a link between cos(2θ) and cos(θ/2). The fourth line is justified by taking the square root of both sides of the equation, given that x� = y implies x = ±√y, while considering the positive and negative roots due to the nature of trigonometric functions in different quadrants.

Step-by-step explanation:

The proof of the cosine half-angle identity starts with the double angle identity for cosine, which is cos(2θ) = cos²θ - sin²θ. This identity can also be written as cos(2θ) = 2cos²θ - 1 or cos(2θ) = 1 - 2sin²θ. The proof seeks to express cos(θ/2) in terms of cos(θ). To arrive at the half-angle identity, we use the following steps:

1. Start with the identity cos(2θ) = 2cos²θ - 1.
2. Add 1 to both sides to get 1 + cos(2θ) = 2cos²θ.
3. Divide both sides by 2, resulting in (1 + cos(2θ)) / 2 = cos²(θ/2). This is known as the half-angle identity for the cosine function.
4. Take the square root of both sides to find cos(θ/2) = ±√((1 + cos(θ)) / 2).

The third line of the proof, where we divide by 2, is the step where we transform the original identity into the half-angle identity. The justification for the fourth line, which involves introducing the square root, is based on the principle that if x² = y, then x = ±√y. In the context of trigonometric identities, we consider both the positive and negative roots because trigonometric functions can have positive or negative values, depending on the angle's quadrant.