Provided that all trigonometric functions below are defined for x, the three pythagorean identities are:
sin2(x)+cos2(x)=1
tan2(x)+1=sec2(x)
1+cot2(x)=csc2(x).
Pythagorean identities.
Sum and Difference Formulas
Recall the sum and difference formulas for sine and cosine:
sin(x+y)=sin(x)cos(y)+cos(x)sin(y)
sin(x−y)=sin(x)cos(y)−cos(x)sin(y)
cos(x+y)=cos(x)cos(y)−sin(x)sin(y)
cos(x−y)=cos(x)cos(y)+sin(x)sin(y).
Sum and difference formulas.
Double Angle Identities
If we substitute y=x into the addition formulas above, we obtain the double angle formulas for sine and cosine. These formulas
are very important in calculus.
sin(2x)=2sin(x)cos(x)
cos(2x)=cos2(x)−sin2(x)=1−2sin2(x)=2cos2(x)−1
cos2x=1+cos(2x)2
sin2x=1−cos(2x)2
Double angle formulas example 1
Double angle formulas example 2