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If sin x - cos x = p ,  then what is cos2 (2x) equal to?
1. p2 - 1
2. p2(2 - p2
3. p2(2 + p2
4. p2 + 1

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Correct Answer - Option 2 : p2(2 - p2

Concept:

sin2 x + cos2 x = 1

sin 2x = 2 sin x cos x

cos 2x = cos2 x - sin2 x

 

Calculation:

Given: sin x - cos x = p

By squaring both the sides, we get

⇒ sin2 x + cos2 x - 2 sin x cos x = (p)2

As we know that, sin2 x + cos2 x = 1 and sin 2x = 2 sin x cos x

⇒ 1 - sin 2x = p2 

⇒ sin 2x = 1 - p2

As we know that, sin2 x + cos2 x = 1

∴ sin2 2x + cos2 2x = 1

⇒ cos2 2x = 1 – sin2 2x

= 1 – (1 - p2)2

= 1 - 1 - p4 + 2p2

= p2(2 - p2

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