Question

(sin2a+sin4a)²+(cos2a+cos4a)²=4cos²a

Answer 1

Expanding this out gives:
$(sin^2 (2a) + 2sin(2a) sin(4a) +sin^2 (4a)) \\ +(cos^2 (2a) + 2cos(2a) cos(4a) +cos^2 (4a)) \\ = 4cos^2 (a)$

Simplify by using pythagorean identity:
sin^2 + cos^2 = 1
$2sin(2a)sin(4a) + 2cos(2a)cos(4a) + 2 = 4cos^2 (a)$

Next use cosine difference identity:
cos(x-y) = cos(x)cos(y) +sin(x)sin(y)
$2 cos(4a-2a) +2 = 4cos^2 (a)$

Next use cosine double angle identity:
cos(2x) = 2cos^2 - 1
$2(2cos^2 (a) -1) + 2 = 4cos^2 (a)$

Finally distribute and Right side  = Left side
$4 cos^2 (a) = 4 cos^2 (a)$
Identity is proved