Simplifying \frac{\sin x+\sin x \tan^2 x}{\tan x} to \sec x

eiraszero11cu

eiraszero11cu

Answered question

2022-01-16

Simplifying sinx+sinxtan2xtanx to secx

Answer & Explanation

Thomas Lynn

Thomas Lynn

Beginner2022-01-17Added 28 answers

In these types of questions, the general method is to convert the expression into sines and cosines. Letting s=sinx,c=cosx, we get
sinx+sinxtanx2tanx=s+s(sc)2sc
=sc2+s3sc
=s2+c2c
=1c=secx
where we used that tanx=sinxcosx=sc and sinx2+cosx2=1 i.e. s2+c2=1
ramirezhereva

ramirezhereva

Beginner2022-01-18Added 28 answers

Using
1+tan2x=cos2x+sin2xcos2x=1cos2x
then
sinx(1+tan2(x))tanx=sinxcos2(x)tan(x)=1cosx

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