Question

Question #75a89

Answer 1

`"see explanation"`

Explanation:

`"using the "color(blue)"trigonometric identities"`

`•color(white)(x)1+tan^2x=sec^2x`

`•color(white)(x)tanx=sinx/cosx`

`•color(white)(x)secx=1/cosx`

`"consider the LHS"`

`rArr(tanx)/(1+tan^2x)`

`=(sinx/cosx)/(sec^2x)`

`=(sinx/cosx)/(1/cos^2x)`

`=sinx/cancel(cosx) xxcancel(cos^2x)^(cosx)`

`=sinxcosx="RHS"rArr"proven"`

Answer 2

See below.

Explanation:

We're trying to prove `tan x / (1+tan^2 x) = sin x cos x`.

Let's manipulate the left side since it's more complicated.

There is a trigonometric identity that states `1 + tan^2 x = sec^2x`. (It's an alternative form of `sin^2x + cos^2x = 1`; simply divide the whole equation by `cos^2x`.)

Thus,

`tan x / (1+tan^2 x)`

`= tan x / (sec^2 x)`

`= (sin x / cos x)/(1/cos^2x)`

`=sin x / cancel(cos x) * cancel(cos^2 x)^cos x /1`

`=sin x cos x`