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in Trigonometry by (45.0k points)
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\(\frac{\text{sin 5x}}{\text{sin x}}\) is equal to

A. 16 cos4 x – 12 cos2 x + 1

B. 16 cos4 x + 12 cos2 x + 1

C. 16 cos4 x – 12 cos2 x - 1

D. 16 cos4 x + 12 cos2 x - 1

1 Answer

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by (47.4k points)
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Best answer

Correct answer is A.

Given \(\frac{\text{sin 5x}}{\text{sin x}}\)

Let 5x = 3x + 2x

Then

[using sin (A+B) = sin A cos B + cos A sin B]

= (3 – 4 sin2x)(2 cos2x -1) + (4 cos3x – 3 cos x)(2 cosx)

= (6 cos2 x – 3 – 8 sin2 x cos2x + 4 sin2 x) + (8 cos4x - 6 cos2x)

[using sin2x + cos2x = 1]

= – 3 – 8 (1 - cos2x) cos2x + 4 (1 - cos2x)+ 8 cos4x

= – 3 – 8 cos2x + 8 cos4x + 4 - 4 cos2x + 8 cos4x

= 16 cos4x – 12 cos2x + 1

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