Question

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

Answer 1

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`