Question

$\bold{Prove \: That \: , \: Cos \: 4x \: = \: 1 - {Sin}^{2}x \: \times {Cos }^{2}}$

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Answer 1

Answer:

cos2(2x)+sin2(2x)=1

Explanation:

Remember the equation cos2x+sin2x=1? 

Well the x refers to any number so if your number is 2x, then cos22x+sin22x=1

You can also prove this by using the double angle formula

cos2(2x)+sin2(2x)

=(cos2x−sin2x)2+(2sinxcosx)2

=cos4x−2sin2xcos2x+sin4x+4sin2xcos2x

=cos4x+2sin2xcos2x+sin4x

=(cos2x+sin2x)2

=12

=1

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Answer 2

Note that cos4x=(cos2x)2

L.H.S.

=cos^4x−sin^4x

=(cos^2x)^2−(sin^2x)^2

=(cos^2x+sin^2x)(cos^2x−sin^2x)

=(cos^2x+sin^2x)(cos^2x+sin^2x−2sin^2x)

=(1)(1−2sin^2x)

=1−2sin^2x

=R.H.S.