Question #75a89

2 Answers
Nov 20, 2017

"see explanation"see explanation

Explanation:

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

•color(white)(x)1+tan^2x=sec^2xx1+tan2x=sec2x

•color(white)(x)tanx=sinx/cosxxtanx=sinxcosx

•color(white)(x)secx=1/cosxxsecx=1cosx

"consider the LHS"consider the LHS

rArr(tanx)/(1+tan^2x)tanx1+tan2x

=(sinx/cosx)/(sec^2x)=sinxcosxsec2x

=(sinx/cosx)/(1/cos^2x)=sinxcosx1cos2x

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

=sinxcosx="RHS"rArr"proven"

Nov 20, 2017

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