to prove: cos4A=1-sin^2A+1/8sin^4A

Question

it is to be proved

Answer 1

First check the supposed identity by plugging in A=π/4. sin(A)=1√2, sin²(A)=½, sin⁴(A)=¼.

cos(4A)=cos(π)=-1.

1-sin²(A)-⅛sin⁴(A)=1-½-¹/₃₂≠-1, so the identity is false.

cos(4A)=1-2sin²(2A)=

1-2(2sin(A)cos(A))²=1-8sin²(A)cos²(A)=

1-8sin²(A)(1-sin²(A))=1-8sin²(A)+8sin⁴(A).

We know this should be -1 when A=π/4. 1-8sin²(A)+8sin⁴(A)=1-4+2=-1, which is correct.

When A=π/6, sin²(A)=¼, sin⁴(A)=¹/₁₆, cos(⅔π)=-½. 1-8sin²(A)+8sin⁴(A)=1-2+½=-½, which is also correct.

Therefore the identity should be cos(4A)=1-8sin²(A)+8sin⁴(A) and not the given supposed identity.