Show the identity can be derived from a sum or a difference identity and the pythagorean identity?

Question

#cos(2a)=1-2sin^2a#

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Answer 1 · 2018-06-03T18:50:14

See explanation

Remember the angle sum identity

#color(blue)(cos(x+y)=cos(x)cos(y)-sin(x)sin(y)#

Now let #color(red)(x=y=a#

#cos(a+a)=cos(a)cos(a)-sin(a)sin(a)#

#=>cos(2a)=cos^2(a)-sin^2(a)#

By the pythagorean trig identity #color(blue)(cos^2(a)=1-sin^2(a)#

#cos(2a)=(1-sin^2(a))-sin^2(a)#

#=>cos(2a)=1-2sin^2(a) larr color(red)"What we wanted to show"#