Question

# If sin x + cos x = t, then sin 3x - cos 3x is equal to

A

3t (2 + 1)

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B

t (2 - 3)

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C

t (2 + 3)

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D

-2t + 1)

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Solution

## The correct option is B t (2 - 3) We have to simplify/modify sin 3x - cos 3x to express in terms of sin x + cos x. We will express them in terms of sin x and cos x sin 3x - cos 3x = 3 sin x - 4 sin3x - 4 cos3 x + 3 cos x = 3 (sin x+ cos x) - 4 (sin3 x + cos3 x) = 3(sin x + cos x) - 4 [ (sinx+cosx)3 - 3 sin x cos x (sin x + cos x)] We know, t2 = (sinx+cosx)2 =1+ 2 sin x cos x ⇒ sin x cos x =t2−12 ⇒ sin 3x - cos 3x = 3t - 4[ t3 - 3 (t2−1)2t] = 3t - 4[ t3 - 3t32 + 3t2 ] = 3t - 4[- t32 + 3t2 ] = 3t + 2t3 - 6t = 2t3 - 3t = t (2t2 - 3 )

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