Question #75a89

2 Answers
Jim G.
Nov 20, 2017

#"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"#

Fleur
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#

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