Question

PROVE THAT 1+COS^2(2X)=2(COS^4(X)+SIN^4(X))

Answer 1

How do I prove 1+cos^22x=2 (cos^4x+sin^4x)?

I think the question is not given properly due to not knowing to use latex. I understand :)

I solved for 1+cos2(x)=2(cos4(x)+sin4(x)), assumed the asker wrote 2 multiple times by mistake.

But I also decided to solve as 1+cos2(2x)=2(cos4(x)+sin4(x)) and realised my second assumption is correct.

You can write this as:

1.[math] 1+\ cos^2{2x}=... [ /math ] ,so your statement would be less complex than it seemed .

Anyway, I'll start with two formulas that are not used much, hence not considered much, in my opinion. And then find sin4(x), cos4(x) from those. I'll go by RHS first:

I) sin2(x)=1−cos(2x)2

if sin4(x)=sin2(x)×sin2(x),

sin4(x)=(1−cos(2x))24

II) cos2(x)=1+cos(2x)2

if cos4(x)=cos2(x)×cos2(x)

cos4(x)=(1+cos(2x))24

RHS:

2(cos4(x)+sin4(x))

=2((1+cos(2x))2+(1−cos(2x))24)

=1+2cos(2x)+cos2(2x)+1−2cos(2x)+cos2(2x)2

=2+2cos2(2x)2

[Math Processing Error]

LHS:

2(cos4(x)+sin4(x))

[Math Processing Error]

=2(1−2sin2(x).cos2(x))

=2−4sin2(x)cos2(x)

=2−4.(1−cos(2x))2.(1+cos(2x))2

=2−(1−cos2(2x))

=2+cos2(2x)−1

[Math Processing Error]