Question

Verify the identity: cos4u = cos^2 (2u) - sin^2 (2u)

Answer 1

Use the trigonometric identity:

$\cos { \left( A+B \right) = } \cos { \left( A \right) } \cos { \left( B \right) } -\sin { \left( A \right) \sin { \left( B \right) } }$

Now, if A=2u and B=2u:

$\cos { \left( 2u+2u \right) } \\ \\ =\cos { \left( 4u \right) } \\ \\ =\cos { \left( 2u \right) } \cos { \left( 2u \right) } -\sin { \left( 2u \right) \sin { \left( 2u \right) } } \\ \\ =\cos ^{ 2 }{ \left( 2u \right) -\sin ^{ 2 }{ \left( 2u \right) } }$