Buktikan identitas berikut

$1. cos ^{4} alpha (1+tan ^{4} alpha ) = 1-2 sin ^{2} alpha cos alpha ^{2}$

Buktikan identitas berikut

$1.cos ^{4} alpha (1 + tan ^{4} alpha )$
= $cos ^{4} alpha + cos ^{4} alpha . tan ^{4} alpha$
= $cos ^{4} alpha + cos ^{4} alpha( frac{sin ^{4} alpha }{cos ^{4} alpha } )$
= $cos ^{4} alpha + sin ^{4} alpha$
= $(cos ^{2} alpha .cos ^{2} alpha ) + (sin ^{2} alpha . sin ^{2} alpha )$
= $[cos ^{2} alpha (1 - sin ^{2} alpha )] + [sin ^{2} alpha (1 - cos ^{2} alpha )]$
= $cos ^{2} alpha - sin ^{2} alpha .cos ^{2} alpha + sin ^{2} alpha - cos ^{2} alpha.sin ^{2} alpha$
= $1 - 2cos ^{2} alpha .sin ^{2} alpha$

TERBUKTI