If cos ⁡ 5 θ = 5 c o s θ ⁡ - 20 cos 3 ⁡ θ + a cos 5 ⁡ θ + b , then the value of a + b is equal to

Question

If cos ⁡ 5 θ = 5 c o s θ ⁡ - 20 cos 3 ⁡ θ + a cos 5 ⁡ θ + b , then the value of a + b is equal to, 20, 16, - 16, 15

MARKS App · Accepted Answer

cos ⁡ 5 θ = cos ⁡ 4 θ + θ = cos ⁡ 4 θ cos ⁡ θ - sin ⁡ 4 θ sin ⁡ θ = 2 2 c o s 2 θ - 1 2 - 1 cos ⁡ θ - 2.2 sin ⁡ θ cos ⁡ θ 2 c o s 2 θ - 1 sin ⁡ θ = cos ⁡ θ 2 4 cos 4 ⁡ θ + 1 - 4 cos 2 ⁡ θ - 1 - 4 sin 2 ⁡ θ cos ⁡ θ 2 cos 2 ⁡ θ - 1 = 8 cos 5 ⁡ θ - 8 cos 3 ⁡ θ + 2 cos ⁡ θ - cos ⁡ θ - 4 1 - cos 2 θ cos ⁡ θ 2 cos 2 ⁡ θ - 1 = 8 cos 5 ⁡ θ - 8 cos 3 ⁡ θ + cos ⁡ θ - 4 cos ⁡ θ 2 cos 2 ⁡ θ - 1 - 2 cos 4 ⁡ θ + cos 2 ⁡ θ = 16 cos 5 ⁡ θ - 20 cos 3 ⁡ θ + 5 cos ⁡ θ Clearly, a = 16 , b = 0

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