How can you show #tan^2 x(1+cos2x)=2sin^2x#?

1 Answer
Geoff K.
Dec 16, 2016

We can use the #color(blue)"double angle identity for cos(2x)"# as well as the #color(green)"ratio identity for tan x"#.

Explanation:

#tan^2 x (1+color(blue)(cos 2x))= tan^2 x (cancel 1 + color(blue)(2cos^2 x - cancel 1))#
#color(white)(tan^2 x (1+cos 2x))= tan^2 x (2 cos^2 x)#
#color(white)(tan^2 x (1+cos 2x))= 2color(green)(tan^2 x) cos^2 x#
#color(white)(tan^2 x (1+cos 2x))= 2color(green)((sin^2 x / cancel(cos^2 x))) cancel(cos^2 x)#
#color(white)(tan^2 x (1+cos 2x))= 2 sin^2 x#

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