Question

show that sin5x cos3x = 1/2 (sin8x +sin 2x)​

Answer 1

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sin5xcos3x=sin6xcos2x x∈[0,π]

⇒

2

1

(sin 8x + sin 2x) =

2

1

(sin 8x + sin 4x)

⇒ sin 2x - sin 4x = 0

⇒-2sin x cos 3x = 0

⇒ sin x = 0 or cos 3x = 0.

That is, x=nπ(n∈I), or 3x=2kπ±π/2(k∈I).

Therefore, since x∈[0,π], the given equation is satisfied if x=0,π,π/6,π/2,5π/6.

Hence, there are 5 solutions

Answer 2

Answer:

sin5xcos3x=sin6xcos2x x∈[0,π]

⇒

2

1

(sin 8x + sin 2x) =

2

1

(sin 8x + sin 4x)

⇒ sin 2x - sin 4x = 0

⇒-2sin x cos 3x = 0

⇒ sin x = 0 or cos 3x = 0.

That is, x=nπ(n∈I), or 3x=2kπ±π/2(k∈I).

Therefore, since x∈[0,π], the given equation is satisfied if x=0,π,π/6,π/2,5π/6.

Hence, there are 5 solutions

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