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Question

What are the formulae of (1)1+cos2x(2)1-cos2x.


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Solution

Step 1. Find the formula for 1+cos2x.

As we know that,

cos(a+b)=cosacosb-sinasinb

Substitute a=b=x in the above equation.

cos(x+x)=cosxcosx-sinxsinxcos2x=cos2x-sin2x

1+cos2x=1+cos2x-sin2x=sin2x+cos2x+cos2x-sin2x[sin2x+cos2x=1]=2cos2x

Thus, 1+cos2x=2cos2x

Step 2. Find the formula for 1-cos2x.

1-cos2x=1-(cos2x-sin2x)=1-cos2x+sin2x=sin2x+cos2x-cos2x+sin2x[sin2x+cos2x=1]=2sin2x

Thus, 1-cos2x=2sin2x

Hence,

1+cos2x=2cos2x1-cos2x=2sin2x


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