Question #545b9

1 Answer
Oct 21, 2016

Answer:

#cos(x)(tan(x)+cot(x))=1/sin(x)#

Explanation:

We will use the definitions of the tangent and cotangent functions:

  • #tan(x) = sin(x)/cos(x)#
  • #cot(x) = cos(x)/sin(x)#

as well as the identity #sin^2(x)+cos^2(x) = 1#
Note that subtracting #sin^2(x)# from both sides produces the form we will use:

  • #cos^2(x) = 1-sin^2(x)#

With those, we have

#cos(x)(tan(x)+cot(x)) = cos(x)(sin(x)/cos(x)+cos(x)/sin(x))#

#=cancel(cos(x))*sin(x)/cancel(cos(x))+cos(x)*cos(x)/sin(x)#

#=sin(x)+cos^2(x)/sin(x)#

#=sin(x)+(1-sin^2(x))/sin(x)#

#=sin(x)+1/sin(x)-(sin(x)cancel(sin(x)))/cancel(sin(x))#

#=cancel(sin(x))+1/sin(x)-cancel(sin(x))#

#=1/sin(x)#