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 (θ/2) = (1 + cosθ)/2 blank line. ________ 5. cosine squared (θ/2) = (1 + cosθ)/2 multiplication 6. cosine (θ/2) = ±√((1 + cosθ)/2) 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 is known as the cosine squared identity, used to derive the half-angle identity. By dividing and rearranging the cosine double-angle identity, we obtain the form that represents the square of the cosine of half the angle. This manipulation establishes the relationship between the cosine of an angle and its half.

Step-by-step explanation:

The question revolves around proving the cosine half-angle identity which is an important concept in trigonometry. When examining the provided proof, the third line of this proof, which results from dividing both sides of the equation by 2, is known as the cosine squared identity. Mathematically, it transforms the original cos(2θ) identity into a form that represents the square of cos(θ/2). The justification for arriving at the fourth line comes from substituting cos(2θ) with the equivalent expression (2cos²θ - 1) and manipulating the equation to isolate cos²(θ/2).

Given the identities such as cos(2θ) = cos²θ - sin²θ, which can further be expressed as cos(2θ) = 2cos²θ - 1 (from the double-angle identities), and sin²θ + cos²θ = 1 (from the Pythagorean identity), we can establish relationships between various trigonometric expressions. The half-angle identities specifically give us a method to find the cosine and sine of half of any given angle, which are particularly useful in solving trigonometric equations and in calculus.