(a)

½ x = a
x = 2a

2•cos(x) = 2•cos(2a)
= 2•cos²(a) - 2•sin²(a)

3 • sin²(a) - sin(a) • cos(a) = cos²(a)
⇨ ÷cos²(a) at both sides

tan²(a) - ⅓ • tan(a) = ⅓
⇨
( tan(a) - (⅙)² ) = ⅓ + (⅙)² = ¹³⁄₃₆

⇨
tan(a) = ⅙ + √(¹³⁄₃₆) ≈ 0.7675919
tan(a) = ⅙ - √(¹³⁄₃₆) ≈ -0.4342585
⇨
a₁ = 37.51° ； a₂ = 217.51°
a₃ = 156.53° ； a₄ = 336.53°
⇨
x₁ = 55.02° ； x₂ = 235.02°
x₃ = 313.06° ； x₄ = 133.06°

(b)

cot(x) = cos(x) / sin(x)

cosec(x) = 1 / sin(x)

× sin(x) at both side of the equation
⇨
2 • cos(x) = 3 • sin(x) + 2
⇨
4 • cos²(x) = 9 • sin²(x) + 12•sin(x) +4

⇨ + 4 • sin²(x) at both sides

13 • sin²(x) + 12•sin(x) = 0
⇨
13 • sin²(x) + 12•sin(x) = 0
sin(x) = ¹²⁄₁₃

x₁ = 67.38°
x₂ = 180° + x₁ = 247.38°

(c)
4•sin²(x) - 2•sin(x)•cos(x)
= sin²(x) + cos²(x)

4•sin²(x)
= sin²(x) + cos²(x) + 2•sin(x)•cos(x)
= [ sin(x) + cos(x) ]²

2•sin(x) = sin(x) + cos(x)
sin(x) = cos(x)
x₁ = 45°
x₂ = 180° + x₁ = 225°

or

-2•sin(x) = sin(x) + cos(x)
-3sin(x) = cos(x)
tan(x) = -⅓
x₁ = 161.565°
x₂ = 180° + x₁ = 341.565°

May I know for (a) where does the 2 cos x go to?