What are the formulae of (1)1+cos2x(2)1-cos2x.
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-sinxsinx⇒cos2x=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
2x and (−12) are ___ of the algebraic expression (2x−12).