5 Use compound and double angles to prove that: \( 5.1 \sin 3 x=3 \sin x-4 \sin ^{3} x \) \( 5.2 \cos 3 x=\cos x\left(1-4 \sin ^{2} x\right) \) \( \frac{5.3}{5.4} \cos 4 x=8 \cos ^{4} x-8 \cos ^{2} x+1 \) \( \frac{5.5}{5} \sin 4 x=\cos 4 x=2 \sin ^{2} 2 x \) 5.6 \( \sin 4 x=4 \sin x \sin x-8 \cos ^{3} x-4 \cos x \sin ^{3} x \)

We will prove each identity step‐by‐step using the compound angle and double angle formulas. ────────────────────────────── Proof of 5.1 ────────────────────────────── We wish to show that   sin 3x = 3 sin x – 4 sin³ x. Step 1. Write 3x as 2x + x and use the sine angle–sum formula:   sin 3x = sin(2x + x) = sin 2x · cos x + cos 2x · sin x. Step 2. Replace sin 2x and cos 2x with their double–angle formulas:   sin 2x = 2 sin x cos x  and  cos 2x = 1 – 2 sin² x. Thus,   sin 3x = (2 sin x cos x) cos x + (1 – 2 sin² x) sin x       = 2 sin x cos² x + sin x – 2 sin³ x. Step 3. Factor sin x:   sin 3x = sin x [2 cos² x + 1 – 2 sin² x]. Step 4. Express cos² x in terms of sin² x (using cos² x = 1 – sin² x):   2 cos² x + 1 – 2 sin² x = 2(1 – sin² x) + 1 – 2 sin² x              = 2 – 2 sin² x + 1 – 2 sin² x              = 3 – 4 sin² x. Thus, we get   sin 3x = sin x (3 – 4 sin² x) = 3 sin x – 4 sin³ x, which is the desired result. ────────────────────────────── Proof of 5.2 ────────────────────────────── We wish to show that   cos 3x = cos x (1 – 4 sin² x). Step 1. Write 3x as 2x + x and use the cosine angle–sum formula:   cos 3x = cos(2x + x) = cos 2x · cos x – sin 2x · sin x. Step 2. Replace cos 2x and sin 2x by their respective double–angle formulas:   cos 2x = 1 – 2 sin² x  and  sin 2x = 2 sin x cos x. Thus,   cos 3x = (1 – 2 sin² x) cos x – (2 sin x cos x) sin x       = cos x – 2 sin² x cos x – 2 sin² x cos x. Step 3. Combine like terms:   cos 3x = cos x – 4 sin² x cos x = cos x (1 – 4 sin² x). ────────────────────────────── Proof of 5.3 and 5.4 ────────────────────────────── We wish to prove that   cos 4x = 8 cos⁴ x – 8 cos² x + 1. Step 1. Write 4x as 2(2x) and use the double–angle formula for cosine:   cos 4x = cos(2·2x) = 2 cos² 2x – 1. Step 2. Express cos 2x in terms of cos x:   cos 2x = 2 cos² x – 1. Then,   cos² 2x = (2 cos² x – 1)² = 4 cos⁴ x – 4 cos² x + 1. Step 3. Substitute into the expression for cos 4x:   cos 4x = 2(4 cos⁴ x – 4 cos² x + 1) – 1       = 8 cos⁴ x – 8 cos² x + 2 – 1       = 8 cos⁴ x – 8 cos² x + 1. ────────────────────────────── Proof of 5.5 ────────────────────────────── Another standard double–angle identity is   sin 4x = 2 sin 2x cos 2x. Step 1. Replace sin 2x and cos 2x by their formulas:   sin 2x = 2 sin x cos x  and  cos 2x = cos² x – sin² x. Then,   sin 4x = 2 (2 sin x cos x)(cos² x – sin² x)       = 4 sin x cos x (cos² x – sin² x). This form is equivalent to an alternate form of sin 4x. ────────────────────────────── Proof of 5.6 ────────────────────────────── It is also common to write the previous result by expanding the product. Notice that   sin 4x = 4 sin x cos x (cos² x – sin² x)       = 4 sin x cos³ x – 4 sin x cos x sin² x. That is,   sin 4x = 4 cos³ x sin x – 4 cos x sin³ x. This is another acceptable form of the identity. ────────────────────────────── Summary of Answers ────────────────────────────── 5.1  sin 3x = 3 sin x – 4 sin³ x 5.2  cos 3x = cos x (1 – 4 sin² x) 5.3/5.4 cos 4x = 8 cos⁴ x – 8 cos² x + 1 5.5  sin 4x = 4 sin x cos x (cos² x – sin² x) 5.6  sin 4x = 4 cos³ x sin x – 4 cos x sin³ x Any answer that is algebraically equivalent to the ones shown is correct.